Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UF00082386/00001
## Material Information- Title:
- A fault tolerant GEQRNS processing element for linear systolic array DSP applications
- Creator:
- Smith, Jeremy C., 1966- (
*Dissertant*) Taylor, Fred J. (*Thesis advisor*) Jullien, Graham A. (*Reviewer*) Law, Mark E. (*Reviewer*) Principe, Jose C. (*Reviewer*) Mair, Bernard A. (*Reviewer*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1994
- Copyright Date:
- 1994
- Language:
- English
- Physical Description:
- x, 120 leaves : ill., photos ; 29 cm.
## Subjects- Subjects / Keywords:
- Delay circuits ( jstor )
Dynamic range ( jstor ) Fault tolerance ( jstor ) Integrated circuits ( jstor ) Latches ( jstor ) Propagation delay ( jstor ) Redundant components ( jstor ) Semiconductor wafers ( jstor ) Shift registers ( jstor ) Signals ( jstor ) Array processors ( lcsh ) Dissertations, Academic -- Electrical Engineering -- UF Electrical Engineering thesis Ph. D Sistolic array circuits ( lcsh ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Abstract:
- In this work the design of a Galois Enhanced Quadratic Residue Number System (GEQRNS) processor is presented, which can be used to construct linear systolic arrays. The processor architecture has been optimized to perform multiply-accumulate type operations on complex operands. The properties of finite fields have been exploited to perform this complex multiplication in a manner which results ina greatly reduced hardware complexity. The processor is shown to have a high degree of tolerance to manufacturing defects and faults which can occur during operation. The combination of these two factors makes this an ideal candidate for array signal processing applications, where hgih complex arithmetic data rates are required. A prototype processing element has been fabricated in 1.5 um. CMOS technology, which is shown to operate at 40 MHz.
- Thesis:
- Thesis (Ph. D.)--University of Florida, 1994.
- Bibliography:
- Includes bibliographical references (leaves 117-119).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Jeremy C. Smith.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Jeremy C. Smith. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 020381363 ( ALEPH )
32490064 ( OCLC ) AKJ5193 ( NOTIS )
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A FAULT TOLERANT GEQRNS PROCESSING ELEMENT FOR LINEAR SYSTOLIC ARRAY DSP APPLICATIONS By JEREMY C. SMITH A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1994 .. Copyright 1994 by Jeremy C. Smith .. ACKNOWLEDGEMENTS I would like to thank my advisor Dr. Fred J. Taylor for providing me with the means and the environment necessary for getting this work done, and for allowing me the freedom to pursue the research directions I saw fit. Special thanks also go to Dr. Graham A. Jullien for serving on my Ph.D committee from such a far distance away. I would also like to thank Dr. Mark E. Law, Dr. Jose C. Principe and Dr. Bernard A. Mair for serving on my committee. I would also like to thank my parents, whose early preparation and dedication have made my accomplishments possible. I would, especially, like to thank Diane for her patience, dedication and love, which were all necessary ingredients for the completion of this dissertation. .. TABLE OF CONTENTS ACKNOWLEDGEMENTS iii LIST OF TABLES vi LIST OF FIGURES vii ABSTRACT x CHAPTERS 1 INTRODUCTION 1 2 THEORY . 2.1 The Quadratic Residue Number System (QRNS) . 2.2 The Galois-Enhanced QRNS (GEQRNS) . 2.3 Dynamic Range . 2.4 Exam ple . . . 3 KEY ARRAY PROCESSOR IMPLEMENTATION ISSUES. . 3.1 Synchronization . 3.1.1 Traditional Approaches . 3.1.2 True Single Phase Clocked Systems . 3.2 Synchronous vs. Asynchronous Systems . 3.3 Fundamental Manufacturing Limitations . 3.3.1 Defect Size Distribution . 3.3.2 Defect Spatial Distribution and Yield Models . . 4 SYSTEM ARCHITECTURE . 4.1 Architectural Overview . 4.2 Multiply Accumulate PE Architecture . 4.2.1 Modulo-P Adders . 4.3 Forward Mapping: Integer to GEQRNS . 4.4 Inverse Mapping: QRNS to Residue . 4.5 Chinese Remainder Theorem . 5 VLSI IMPLEMENTATION OF PROCESSING ELEMENT . 4 4 6 7 9 15 15 . 16 . 24 . 29 . 33 . 34 . 38 . 46 . 46 . 46 . 50 . 52 . 54 . 54 . 57 5.1 True Single Phase Clocking Scheme .. 5.1.1 Pipeline Registers. . . . .58 5.1.2 Data Storage Shift Register. . . . .59 5.2 Exponentiation ROM. . . . . .60 5.3 Electronic Reconfiguration Switches . . . .65 5.4 PE Performance. . . . . .67 5.5 Early Versions of the Processing element. . . .73 5.5.1 Version One . . . . .73 5.5.2 Version Two. . . . . .74 6 YIELD ENHANCEMENT AND FAULT TOLERANCE. . .79 6.1 Yield Enhancement via Reconfiguration . . .79 6.1.1 Yield Estimates . . .*82 6.2 A Comparison with Replacing Moduli. . . .87 6.3 Detecting Faults. . . . . .90 7 CONCLUSIONS AND FUTURE WORK . . . .95 APPENDICES A VLSI CELL LAYOUTS. . . . . .100 B OBSERVED CHIP DATA. . . . . .109 C COMPUTER PROGRAMS . . . . .110 REFERENCES. . . . . . . .117 BIOGRAPHICAL SKETCH. . . . . .120 .. LIST OF TABLES 2.1 Table of Maximum Dynamic Ranges for Eight to Four Modulus System. 9 2.2 Table of Maximum Inner Product Lengths 9 2.3 Log-Antilog Table for p, = 5 . 11 2.4 Log-Antilog Table for P2 = 13 11 4.1 Full Adder truth table 51 6.1 Table of System Component Areas 83 6.2 Table of Non-Redundant Chip Yields 83 6.3 Table of Redundant Chip Yields 85 .. LIST OF FIGURES 3.1 Single Phase Latch System.17 3.2 Double-latch non-overlapping pseudo two phase system. . .19 3.3 High performance non-overlapping pseudo two phase system. .20 3.4 Delay transformation model . . . . .21 3.5 Transparency problem introduced by complementary phase of clock. .23 3.6 System model for TSPC edge based clocking. . . .25 3.7 Timing waveforms for TSPC edge based clocking . . .26 3.8 System model for TSPC latch based clocking. . . .27 3.9 Timing waveforms for TSPC latch based clocking. . . .28 3.10 Potential skew hazard with TSPC scheme [44] . . . .30 3.11 Clocking against the data flow to exploit skew [44]. . . .30 3.12 Non-local communication problem solution [44]. . . .31 3.13 SEM photographs of defects from early PE fabrications . .35 3.14 Defect density vs. defect radius. . . . .36 3.15 Critical area for Parallel Conductors . . . .38 4.1 System Architecture. . . . . .47 4.2 Processor Architecture . . . . .49 4.3 Standard Modulo P Architecture . . . .51 4.4 Modulo-P Adder Building Block Primitives . . . .52 4.5 Carry Select Modulo Adder . . . . .53 4.6 Forward-mapping (0) conversion module with GEQRNS log table .54 .. 4.7 Inverse-mapping (0-') conversion module . . . . . . . . 55 4.8 CRT block diagram . . . . . . . . . . . . . . 56 5.1 TSPC Pipeline Register . . . . . . . . . . . . 58 5.2 SPICE simulation of fast transition path for shift register . . . 61 5.3 TSPC Shift Register Cell with Storage . . . . . . . . 62 5.4 Floorplan of Exponentiation ROM . . . . . . . . . . 63 5.5 Key ROM Circuit Elements . . . . . . . . . . . . 63 5.6 SPICE simulation of ROM operation . . . . . . . . . 66 5.7 Die photograph of processor . . . . . . . . . . . . 69 5.8 Oscilloscope photo of clock signal and output bit zero . . . . 70 5.9 Pass-through test . . . . . . . . . . . . . . . 71 5.10 Oscilloscope photo of pass-through test output . . . . . . 72 5.11 SPICE simulation of pass-through test . . . . . . . . . 72 5.12 Processor architecture of first version . . . . . . . . . 75 5.13 Die photograph of first version of PE . . . . . . . . . 76 5.14 Oscilloscope photo of non-overlapping clocks for first chip . . . 77 5.15 Processor architecture of second version . . . . . . . . 77 5.16 Die photograph of second version of PE . . . . . . . . 78 6.1 Yield Curves for Various Length Arrays (Scheme 1) . . . . . 86 6.2 Yield Curves for Various Length Arrays (Scheme 2) . . . . . 91 6.3 System Areas for Scheme I (lower) and Scheme 2 (higher) . . . 91 6.4 Adjusted Yield Curves For Scheme I (Best Case) . . . . . 92 6.5 Adjusted Yield Curves For Scheme 1 (Worst Case) . . . . . 92 6.6 Adjusted Yield Curves For Scheme 2 (Best Case) . . . . . 93 6.7 Adjusted Yield Curves For Scheme 2 (Worst Case) . . . . . 93 A.1 ROM Word-line Decoder Cell . . . . . . . . . . . 101 .. A.2 ROM Sense Amplifier 102 A.3 ROM Programming Matrix 103 A.4 Mod P Building Block (Zero) . 104 A.5 Mod P Building Block (One) 105 A.6 DSSR Cell . . 106 A.7 Pipeline Register with Direction Logic 107 A.8 Reconfigurable Switch Element . 108 .. Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy A FAULT TOLERANT GEQRNS PROCESSING ELEMENT FOR LINEAR SYSTOLIC ARRAY DSP APPLICATIONS By Jeremy C. Smith August 1994 Chairman:, Dr. Fred J. Taylor Major Department: Electrical Engineering In this work the design of a Galois Enhanced Quadratic Residue Number System (GEQRNS) processor is presented, which can be used to construct linear systolic arrays. The processor architecture has been optimized to perform multiplyaccumulate type operations on complex operands. The properties of finite fields have been exploited to perform this complex multiplication in a manner which results in greatly reduced hardware complexity. The processor is also shown to have a high degree of tolerance to manufacturing defects and faults which can occur during operation. The combination of these two factors makes this an ideal candidate for array signal processing applications, where high complex arithmetic data rates are required. A prototype processing element has been fabricated in 1.5 Ptm CMOS technology, which is shown to operate at 40 MHz. .. CHAPTER 1 INTRODUCTION Arithmetic bandwidth continues to remain the principal limitation in high speed Digital Signal Processing (DSP) applications. In the past, systolic arrays have been proposed as a means to achieve high computational throughput for compute bound applications [14]. The highly modular nature of systolic arrays makes them attractive for larger than conventional levels of integration such as Ultra Large Scale Integration (ULSI), and Wafer Scale Integration (WSI). In an array processor, each building-block cell or Processing Element (PE) performs some basic arithmetic operation on data which arrives at its inputs. The data flow is in some predetermined and regular ordering, so that operands arrive at the correct processor at the correct time. The architecture of each PE is highly optimized and is then repeated over a large silicon area. In order to realize large monolithic arrays of processing elements, it is necessa ry to cope with the fact that many of the processors in an arbitrary array will have some fatal defect at the time of manufacture. Additionally, as chip transistor count and scaling density increases, so does the probability of operational failure, due to biasrelated physical phenomena [8]. In this work the design of a Residue Number System (RNS) processor is presented, which is used to construct a linear systolic array. The processor architecture has been optimized to perform multiply-accumulate type operations on complex operands. The properties of finite fields have been exploited to perform this complex multiplication in a manner which results in greatly reduced hardware complexity. .. The processor is also shown to have a high degree of tolerance to manufacturing defects and faults which occur during operation. The combination of these two factors makes this an ideal candidate for signal processing applications, where high complex arithmetic data rates are required. In section two, an introduction to the theory leading up to the processor architecture is presented. It will be shown how complex multiplication can be performed in two non-communicating parallel channels via the Quadratic Residue Number System (QRNS) mapping. Furthermore, the mapping may be taken to one more level, where the actual multiplication is performed as a sum of two number-theoretic exponents. Next, some bounds are given on the maximum length of a complex inner product that can be computed, within the given dynamic range of the system. Finally, the chapter concludes with an illustrative numerical example of the mapping techniques. Some key implementation issues which relate to the design of large area integrated circuits (ICs) are presented in Chapter Three. In a broad sense, these are synchronization and manufacturing yield. The first section of Chapter Three describes the clocking techniques that have been historically used in integrated circuits and some of their limitations. The section concludes with the introduction of the true single phase clocked (TSPC) technique, which represents the latest development in synchronous system design. The second section of Chapter Three presents the key issues of IC manufacturing. A physical notion of manufacturing defects will be developed along with some models used to predict integrated circuit yield. An overview of the proposed system will be presented in Chapter Four, along with the architectural design of the processing element. The design tradeoffs which resulted in the final PE implementation are discussed in detail. Finally, the architecture of the support modules necessary to perform the forward and reverse mappings of the input and output data are presented. It is shown that these conversion elements can be implemented exclusively, with the key modules of the PE. .. The VLSI design of the PE is presented in Chapter Five. The chapter focuses heavily on the transistor-level design of the PE. The internal details of each major module of the PE are presented along with computer simulation of their behavior. Some results of the testing of fabricated chips are then presented. It is shown that the PE is capable of maintaining a very high data rate, which is due, ultimately, to the aggressive design techniques used for its electronics. Finally, a discussion of the earlier versions of the PE is presented, which detail the evolution and optimization of the current architecture. An analysis of the fault-tolerant properties of the proposed system is given in Chapter Six. It is shown that the yield of a large-area sixteen processor array can be significantly increased by the chosen redundancy scheme. This scheme is then compared to one which is traditionally used for RNS systems and is shown to be far more efficient and beneficial. Finally, Chapter Seven summarizes the accomplishments of this dissertation. Some concluding remarks and future directions which this work might take, are also presented. .. CHAPTER 2 THEORY 2.1 The Quadratic Residue Number System (QRNS) The Residue Numbering System (RNS) has long been proposed as a means of achieving high-computational bandwidths in signal processing systems [34, 26]. The RNS gets its speed advantage because computations over a large base ring can be implemented over smaller computation rings, due to an isomorphism between elements in the base ring and the direct sum of the computation rings. An integer X in the RNS is represented by an L-tuple of residues: X = (Xl,.,XL) (2.1) where xi =< X >mi is the ith residue and mi is the ith modulus. The production rule for the ith digit in a RNS computation is zi = < Xi + Yi >m, (2.2) zi = < Xi X Yi >mi which represents modular addition and multiplication, respectively. The importance of Equation 2.2 is that the computation of any digit in the L-tuple is independent of any other digit. This means that there are no carries between the residue channels. If the residue channels are of small wordwidth, then high computation rates can be achieved in physical systems. This is the central theme in RNS implementations. The rule which maps computations over the implementation rings back to the base ring is the Chinese Remainder Theorem (CRT). The CRT is given below: .. X fL X = < rhi x < (ri)-1 Xi >m > M (mod M) (2.3) Where M = ill mi, and for i,j E {1,2,3,.,L}, gcd(mi, mj) = 1 for i 5j, and rhi = M/mi, with rhi x (nrhi)-1 = 1. A historical complaint about RNS systems is that the speed gains obtained by the fast parallel channels are lost in the CRT since it requires a final mod(M) operation across the entire dynamic range. However, this is less of an issue today as large wordwidth fast binary adders are regularly demonstrated in the literature [23, 27]. As we will see shortly, smaller binary adders can be used to make larger modulo adders, with minimal area complexity. Complex operations in the RNS can be performed by simply emulating conventional architectures with RNS elements (Complex Residue Numbering System (CRNS)). However, the QRNS first introduced by Leung [17] and later developed by Krogmeier and Jenkins [12] is a much better way of performing complex operations. In QRNS real and imaginary components are encoded into two independent quantities, whereby complex operations can be performed independently in two parallel channels. This requires that the moduli be restricted to primes of the form 4k + 1. If this is so, the equation x2 + 1 = 0 (mod p) (2.4) has two solutions in the ring Zp, denoted by 3 and 3-1, which are additive and multiplicative inverses of each other. We define a forward mapping 0: Zp[j]/(j 2 + 1) --+ ZP x Zp to be 0(a + jb) = (z,z*) (2.5) Z (2.5) z < a +b >P z*- < a -b >P We will call the z and z* operands the normal and conjugate components, respectively. The inverse mapping 0-': Z, x ZP Zp[j]/(j2 + 1) is given by .. 0-I(z,z*) =< 2-1(z + z*) >p +j < 2-1-1(z z*) >p (2.6) If (z, z*), (w, w*) E Z, x Z,, then addition and multiplication operations in the ring < Zp x Zp,, +,. > are given by (z,z*) + (w, w*) = (z + w, z* + w*) (2.7) (z, z*) (w, w*)= (zw, z*w*) Since the z and z* channels are independent, they can be implemented in two separate channels. Complex arithmetic can thus be performed in two simultaneous operations, executed in one clock cycle. 2.2 The Galois-Enhanced QRNS (GEQRNS) The properties of Galois fields can be used to further simplify complex multiplication in the RNS [25, 24, 18, 35]. It is well known that for any prime modulus p that there exists some a E Zp that generates all non-zero elements of the field GF(p). That is, any non-zero element in Zp can be represented by ak, where k E {0, 1,2,. ,p 2}. Since we can represent all elements of GF(p) {0} by exponents, multiplication can be performed via exponent addition. This is highly desirable from a hardware standpoint since n-bit adders tend to be smaller and faster than n-bit multipliers. A number theoretic logarithm table is used to obtain the power of a for each QRNS operand, and an antilogarithm table is used to recover the summed powers (modulo p-i). Exploitation of this cyclic property also permits the use of moduli which are larger than those typically used for RNS systems (typically less than five bits), since a hardware multiplier is not needed. This translates into increased dynamic range for fewer channels. Eight bit moduli have been used in this implementation, but the technique could easily be extended to 10 or 11 bit moduli. Beyond this, the logarithm and antilogarithm tables become too large and slow to be of beneficial use. .. I -2.3 Dynamic Range The legitimate range of integers in a RNS system is [0, M 1] (where M II1 Ps). All 4k + 1 primes bounded by eight bits belong to the set { 241, 233, 197, 181, 173, 157, 149, 137, 113, 109, 101, 97, 89, 73, 61, 57, 53, 41, 37, 29, 17, 13, 5 }. In theory, a RNS system could be constructed in which the maximum range was the product of all of these primes 7.796643721 x 1042( [2142J). Clearly this would result in an impractical implementation, as a massive CRT would be needed to recover the residues. Systems of practical interest would constitute moduli sets of say the first eight moduli. If a signed representation for integers is desired, we can divide the interval [0, M) up evenly into a positive half and a negative half. Since M will be odd for any product of moduli we have defined, the dynamic range will be [-(M 1)/2, (M 1)/2]. Each integer X is mapped onto the range [0, M 1] according to =X (mod pi) X > 0 x = Ixx2.,L = -~p-X (odp) ~ (2.8) A (X (mod pi)) X< 0 where, X mod(pi) is the least positive residue of X with respect to pi. The negative part of the dynamic range thus maps to the upper part of the legitimate range: PositiveRange : [0, (M 1)/2] (2.9) NegativeRange : [(M + 1)/2, M 1] Table 2.1 depicts the maximum ranges achievable for eight through four modulus systems. We will consider the four modulus case. We are interested in performing inner product computations of the form: N-1 c = 1 a(k)b(k) (2.10) k=O .. where a(k) and b(k) are complex sequences. We will assume that the real and imaginary parts of both sequences are signed numbers where -2' + 1 < a, ai 5 2' 1 (2.11) -20 + 1 < b ,bi < 20 I The real and imaginary components of Equation 2.10 will thus satisfy the bounds 2N(2a 1)(20 1) < c, _< 2N(2' 1)(21' 1) (2.12) We must now contain these limits within our total dynamic range so that overflow will not occur during the computation of Equation 2.10, we thus obtain the following inequality (M- 1) > 2N(2c_ 1)(2# 1) (2.13) 2 which implies that the maximum inner product length NMAX is M-1 NMAX <_ 4(2c 1)(20 1) (2.14) Some tabulated inner product lengths are shown in Table 2.2 for various values of a and /3. We will assume that integers with a,/3 > 7 represent quantities which are known a priori, as it is unlikely that numbers of larger wordwidths would be available from high-speed signal acquisition circuits. Large wordwidth quantities would typically be filter coefficients, which can be reduced modulo pi by a host processor, prior to entering the RNS system. Thus, two random eight-bit data streams, or one random eight-bit and one pre-known data stream of wordwidth greater than eight bits can be input. .. Table 2.1: Table of Maximum Dynamic Ranges for Eight to Four Modulus System. Dynamic Range Modulus Set H- Closest power of 2 241,233,197,181,173,157,149,137 1110121095908704853 [259J 241,233,197,181,173,157,149 8103073692764269 L252] 241,233,197,181,173,157 54383044917881 L245J 241,233,197,181,173 346388821133 [238] 241,233,197,181 2002247521 L2 30 Table 2.2: Table of Maximum Inner Product Lengths Maximum Inner Product Length a # NMAX 7 7 31,034 7 9 7,713 7 11 1,925 7 13 481 7 15 120 2.4 Example We will now consider a sample calculation based on the theory presented so far. For this simple case, the smallest two 4k + 1 primes will be used as moduli. These are pi = 5 and P2 = 13. Some preliminary constants that will be needed by Equation 2.3, Equation 2.5 and Equation 2.6 will be presented first, as well as the contents of the lookup tables needed to compute the number theoretic logarithm and antilogarithm values. For the the CRT (Equation 2.3) we will need to know the values of M, fil, (rl)-, rh2 and (rn2)-1. These are obtained as follows: .. M =Imi = M= 5 x 13 = 65 1 = rhi =13 mi2 513 135 rn2=5 < 7-2 X (772)_ >13 =-- 1 = (rtI2)-1 -- 8 The forward QRNS mapping (Equation 2.5) requires that we find the two constants such that the equation < x2 + 1 0 >P can be solved. These are obtained for the moduli used here as follows: 1=: f12 + 1=3 (2.16) 2 +1 013 = 325 32 -8 We note that <2x3>5 = l and that <2+3>5 = 0, thus 2 and 3 are multiplicative and additive inverses of each other, modulo 5. Similarly,-< 5 x 8 >13 = 1 and <5 + 8 >13 = 0. Thus, two elements always exist in each case, which behave exactly like the imaginary operators (j) which we are familiar with. The inverse QRNS mapping requires that we also find the multiplicative inverse of 2, modulo our prime moduli. These are: < 2-1 >5 3 (2.17) < 2- >13 7 Finally, we need to obtain the logarithm-antilogarithm tables to perform the QRNS to GEQRNS mapping and the GEQRNS to QRNS inverse mapping, respectively. We recall that we can only generate the non-zero elements in GF(p) with some ak (where k is modulo p 1). We must thus consider a zero as a special case, which will be denoted by *. The tables are given below, with the generators for each case. The number theoretic logarithm is obtained from going right-to-left in the tables, and the anti-logarithm, from left-to-right. .. Table 2.3: Log-Antilog Table for pi = 5. Log-Antilog Table for pi = 5, a = 3 Power < a 0 < 3<0>4 >5 1 1 < 3<1>4 >5 3 2 < 3<2>4 >5 4 3 < 3<3>4 >5 2 0 Table 2.4: Log-Antilog Table for p2 = 13. .. Now, suppose we wish to compute the product of two complex numbers 6 +j3 and 4 + j5. The product using standard arithmetic is 9 + j42. Let us now compute the product using RNS. We must first perform the forward QRNS mapping: 0(6 +j3) (z, z*) z (<6+f x3>5,<6+2x3>13) = (2,8) Z* (<6-f J1 x3 ,<6-2 x3>13) = (0,4) 0(4 +j5) (w,w*) (2.18) w (<4+fJX5>5,< 4-+-f2x5>13) = (4,3) w* = (<4-f X5>5,<4-f2x5>13) = (4,5) Thus, our complex numbers map into the set of ordered pairs 6+j3 *- (2,8)(0,4) 4+j5 4-4 (4,3)(4,5) (2.19) At this point we can perform the multiplication as an actual multiplication as in QRNS, or we can use logarithmic addition as in GEQRNS. For now we will use QRNS. Performing component wise multiplication as usual, we obtain (2,8)(0,4) X (4,3)(4,5) 4 (2.20) (< 2 x 4 >5, < 8 x 3 >13)(< 0 x 4 >5, < 4 x 5 >13) (3,11)(0,7) We must now use the inverse QRNS mapping, which yields 0-11(3,0) = < (2-1(3 + 0) +j2-1(J1-(3 -0) >, = <3x3+j3x3x3>5 = 4+j2 (2.21) 0-12(11,7) = < (2-1(11 + 7) + j2-1(f2)-1(11 7) >13 = <7x18+j7x8x4>13 = 9+j3 .. Finally, we must use a real and an imaginary CRT to recover our standard integer representation. The following expressions result R < < rhx< 1- x4 >5 >65 + = <<13x<2x4>5>65+<5x<8X9>3>65>65 = <39+35>65 = 9 a= < < rhx < ril- x 2 >5 >65 + < rA2 < r42- x 3 >13 >65 >65 (2.22) S< <13 x < 2 x 2 >5 >65 +<5x<8x3>3>65>65 = <52+55>65 =42 Thus, we obtain the same product as before. Now, we will perform the multiplication in Equation 2.20 using our GEQRNS logarithm tables. We will simply use Table 2.3 and Table 2.4 to lookup the logarithm and antilogarithms of the operands. Thus, (2, 8)(0,4) x (4,3)(4,5) 4 (3 3, 79) (*, 710) x (32, 7s)(32, 73) (2.23) 4 (3<3+2>4, 7<9+8>12)(*, 7<10+3>12) 4 (31, 7-) (*, 71) 4 (3,11)(0,7) We obtain the same result as we did in Equation 2.20, and thus, if we proceed with the inverse QRNS mapping and CRT, will obtain the same final result. Finally, we point out that the GEQRNS is only defined for multiplication. As we will see later, this encoding is highly desirable from a hardware standpoint. In Chapters Four and Five, the architecture of the multiply-accumulate processor is presented. Operands are input to the multiplier portion of the chip as GEQRNS exponents. Once they .. 14 are multiplied, their product is converted back to QRNS inside the processor and subsequently used in the accumulate sections of the chip. .. CHAPTER KEY ARRAY PROCESSOR IMPLEMENTATION ISSUES 3.1 Synchronization The continual increase in integrated circuit scaling density has permitted the development of chips which simultaneously exhibit increased complexity and operating speed. It is now possible to build entire systems on a single chip which consist of many interacting circuit elements. By far, the largest gains in integration are attained with monolithic (single chip) ICs. This is because the delay times associated with communication between modules in chip can easily be an order of magnitude less than those associated with chip-to-chip communication. Synchronous systems constitute the bulk of chips fabricated today. A synchronous system is one in which data is passed to or from communicating modules in a chip on the active "edges" or "states" of a global clock. This greatly simplifies the design of individual circuit elements as data only needs to be stable at these edges or states, rather than in continuous time. As die areas and clock speeds continue to grow, however, it becomes more and more difficult to guarantee that global clock signals arrive at different locations on a chip at the same time. The differences in the arrival times of global clocking signals are due to differences in the path lengths to individual modules. Even if these path lengths can be physically made the same, there are random unavoidable variations in the delay characteristics of the paths that are intrinsic to the manufacturing process. Additionally, the time delays of the clock paths are influenced by thermal and power supply variations, which can also be random in nature or dependent on operating conditions. The difference between arrival times of the clock signal is known as clock .. skew. Clock synchronization is a nontrivial problem for present-day large area, high performance die [6]. Clock skew is a fundamental limitation to the maximum speed achievable for very fast chips. As the cycle times of these chips decreases, the skew time becomes more and more of a significant fraction of the total cycle time. The problem is particularly aggravated when data must be passed to and from distant modules in a chip. The choice of a clocking strategy is thus of paramount importance for the design of high performance integrated circuits. The general trend with time has been a reduction in the number of clock signals that are generated and routed around a chip. There are associated tradeoffs, however, in circuit complexity, speed and clocking safety. The sections that follow will describe some of the more modern clocking schemes that have been used successfully in integrated circuits in the past and some emerging technologies for very high performance chips. 3.1.1 Traditional Approaches Pseudo Single Phase Latch Based Systems The simplest type of data storage element is a latch. Latches are used toehold data at the inputs and outputs of combinational logic gates. In is simplest form, a latch passes data present at its input to its output when the clock is high (active). This is the transparent phase. If the data changes at the input of the latch when the clock is high, it will also change at the output after the time it takes for the change to propagate through the latch. When the clock goes low, the data that was present at the falling edge of the clock is stored and cannot change. This is the nontransparent phase. Latches exhibit the lowest transistor count, due to their simplicity. This factor motivates their use in VLSI systems. In later sections, we will expand our definitions of transparent and non-transparent latch phases. .. Inputs Output -*Li 0 LICombinational Next Logic Stage T Li (D. Present Next State State (FD Figure 3.1: Single Phase Latch System Signals propagate through combinational logic networks at different rates, depending on their values. This is due to the internal characteristics of the transistor switching paths comprising the gates. It is not uncommon, for example, to find propagation delays for highs to be twice as long as those for lows, or vice versa. This difference in delay is also related to the function being implemented and to combinations of the input variables. Consider the implementation of a state machine shown in Figure 3.1, which employs a pseudo single phase latch clocking scheme. The output to the next stage and the next state information, are computed from the present inputs and present state data. New data is sampled by the logic network just after the rising edge of the clock, is modified by the combinational logic (CL) and should be ready to be passed on the the next stage (and feedback path) just before the next rising edge of the clock. We are interested on placing constraints on the allowable delays in the logic so that the maximum operating frequency can be obtained. The clock period, ro, is given by the sum of the high and low phases as ro= TrH -I-rL. We desire the clocking to be data independent, which requires that the slowest delay in the logic be less than r,6. At the .. same time we require the fastest delay in the logic to be slower than TH, otherwise a potential conflict could occur. If the fast path delay was significantly faster than -rH then the newly generated data from the CL network could race through the feedback path and to the next stage, thereby corrupting the previously generated (valid) data. Non-deterministic behavior of the system results is this phenomenon occurs. This is called a race condition which must be avoided at all costs. The delay of the CL network, TCL, is thus subject to the two-sided constraint: TH < TCL < 7* (3.1) To put it succinctly, we require that the slow path be fast enough and the fast path be slow enough. This two-sided requirement is very difficult to guarantee in VLSI systems, in the context of process parameters which have some statistical distribution. This clocking scheme was used in some early VLSI chips, but was later abandoned due to its implicit hazards. More security can be obtained by using a multiphase scheme. The tradeoff being, that the simplicity of this scheme is forfeited for reduced risk and ease of design. Non-overlapping Pseudo Two Phase Clocking Non-overlapping pseudo two phase clocking (NPTC) has been the mainstay of the semiconductor industry for several years now. Most integrated circuits designed today, still employ a NPTC scheme. Its popularity has been due to its relative safety and immunity to race conditions. This security is achieved by the introduction of a second clock phase. The active phase of the second clock does not overlap with the active phase of the first clock. There is thus no possibility of a race condition occuring. This is shown in Figure 3.2. Here, a double latch scheme is employed. New input data is let in to the Li latch just after the rising edge of 01. On the falling edge of qS1 it is "frozen" and cannot change again. After time TA, q52 becomes active and the new data propagates to the .. Inputs Li .)12 Outputs LI --. L21 Combinational Logic 01 02 01 02 Next Stage L2- L1 4 Present Next State State TT (2 O Figure 3.2: Double-latch non-overlapping pseudo two phase system CL network. We notice that the fast path race condition has been eliminated, as even if the CL network is very fast for a particular input (i.e. would have computed the new output with a delay less than rj), it could not race to the next stage (or through the feedback path) since the L2 latches are non-transparent. A two-sided constraint has now been reduced to a one-sided constraint. Now, the only requirement of the CL network is that it meet the upper bound on the slowest transition path. This is satisfied if the following equation holds true: TCL < 2 -+ TB + T1 (3.2) Or TCL < To TA where we define To = ro, = T2 = rl + TA + T + TB. We notice that the TA time delay is an overhead that must paid in every cycle. This is the price we have paid for timing safety, along with the increased complexity of the double latches and wiring of the second phase. We also notice that we have effectively made an edge-triggered latch from our cascade of Li and L2 latches, as new data enters the system on the rising edge of 2. This idea will be important later, and we will call an edge sensitive latch a Flip-flop (FF). Flip-flops can be both negative and positive .. ID2 (DI Figure 3.3: High performance non-overlapping pseudo two phase system edge sensitive. The time wasted during 7"A can be gained back if the CL network can be partitioned appropriately. This is shown in Figure 3.3. Here, combinational logic has been placed between latches, rather than after a cascade of two latches. The advantage of this scheme can be appreciated by examining the constraint equations, which can be obtained from a similar analysis: 'TCL1 < T1 TA + 72 TCL2 < 72 + TB+T1 (3.3) The sum of the CL network delays must be less than the clock period, or TCLl + TCL2 < rT. Any combination of delay times that satisfies this constraint can be implemented. If one of the CL networks is faster than the other, we can thus trade delay giving the slow one more time and the fast one less time. The total delay, TCL1 + TCL2, now approaches the clock period, which is more efficient than before. The NPTC clocking scheme is safe as long as the "dead-time" between the phases can be maintained. In the presence of clock skew, however, this requirement .. 012 012 >20 I __C > -d 011 CLI Figure 3.4: Delay transformation model may be violated. For example, if two non-local modules are communicating, then the relative differences in path delays may degrade our safety margin enough for to cause timing errors. We can model the effects of skew by introducing a delay transformation that preserves the overall timing scheme presented so far. For a general logic module shown in Figure 3.4, if we add a positive delay to the all inputs and subtract this delay from all outputs, the overall timing remains the same. We can thus model the effects of skew by considering an overall timing ioop. A circuit with known delays through the combinational logic and uncertain clock skew can be transformed into a system with uncertain delays through the combinational network and no clock skew [7]. Delay is thus added to one piece of the combinational network and subtracted from the other. This delay may be positive or negative. Negative delay has no physical significance in a real system. Negative delays have significance in the transformed system, however. We note that if q$1 is delayed more than q$2 such that this delay is greater than of equal to 7-A an overlap of active phases results, which requires twosided constraints as in the case of the single phase latch clocking. The system is likely to fail at this point. Pseudo Single Phase Edge Clocking If edge triggering is used for the system in Figure 3.1 then the drawbacks of using an additional clock phase would be avoided. Data would move to the next .. adjacent stage (and to the feedback path) on the rising edge of the clock and thus, would not be subject to the lower bound on logic delay as before. Of course the system would still be vulnerable to clock skew between communicating non-local modules. Single phase edge based clocking and NPTC schemes have been used successfully in integrated circuits, with the former associated with higher performance. At this point, the reader is probably wondering why the word "pseudo" appears in all of the descriptions. This is because we have purposefully stayed away from the internal details of the latches. In reality, complementary (inverted) phases of the clock signals must be generated to make all of our latches work. This is due to the nature of CMOS logic which employs two different species of transistors to pass the full range of logic values. If we view the transistors as ideal switches, then their operation (in the most elementary form) requires that a high voltage turns the NMOS device on and the PMOS device off. The converse is true, that a low turns the NMOS off and the PMOS on. The NMOS device will pass a strong zero and a weak one, while the PMOS will pass a strong one and a weak zero. The simplest switch that will pass the full logic level range is a parallel combination of an NMOS and PMOS device, called a transmission gate. This is shown in Figure 3.5, where a simple positive edge-triggered latch is shown. The circuit works by letting the new data into the first half of the latch during the low phase of q$ and presenting it to the output during the high phase of q$. The internal states in the latch are stored on the parasitic capacitances at the inputs of the inverters in the latch (dynamic logic). If the switching time of 0 is zero as in the ideal case, then there is no potential race condition. In reality, however, this time cannot be zero and the clock signal must have a finite slope. There is thus a built-in transparency during the interval r. If the propagation delay through the latch is on the order of r then a race condition exists. We must thus keep r as small as possible, which implies that the clock edge-rate must be high. To gain an appreciation of the problem, suppose that we are working with .. (D in out OA B Ideal case (D (D D Finite transition time (D Finite transition time with inversion delay Figure 3.5: Transparency problem introduced by complementary phase of clock. a typical 5V chip, and we require that 7- < 2ns. The clock edge rate must thus be greater than 2.5 billion volts per second! Of course the output of the latch cannot rise in zero time (nor can the inverters in the latch), so this will relax our delay constraint somewhat. Nevertheless, the transition time of the clock and its complement must be very small. The problem is compounded since is typically generated locally from 0. This implies that will always lag behind 0 since it cannot be produced in zero time, which increases r. There are thus built-in problems associated with inverting the clock signal for a complementary phase. If we had a family of latch circuits that could operate with only one clock phase then this problem would be solved. The edge rate sensitivity will be a fundamental limitation, if the logic block that follows the latches can switch in times on the order of the clock transition time. For very high performance chips this is in fact the case, and very careful attention must be paid to the worst case clock edge rate. .. 3.1.2 True Single Phase Clocked Systems The True-Single-Phase-Clocking (TSPC) scheme [43, 44, 1] represents the state of the art in integrated circuit clocking. In TSPC, the clock signal is never inverted to produce a complementary phase, which significantly ameliorates the problems pointed out earlier. TSPC schemes become more attractive as chip die areas grow towards ULSI and WSI dimensions, as only one clock line needs to be routed around the chip and since clock skew between phases is eliminated. It also much easier to guarantee the long term reliability of a single clock interconnect, or distribution network. Reducing the clocking complexity of the storage elements is important for large systems since the clock capacitive load influences the overall system speed. At the heart of the technique is the use of two types of latch circuits which are alternately transparent on either the high or low phases of the clock. The latch which is transparent on the high phase of the clock and is non-transparent on the low phase is called the N-latch. Likewise, the latch transparent on the low phase and non-transparent on the high phase is called the P-latch. The latches permit combinational logic circuits to be placed between N and P latch sections, or actually embedded in the latches themselves. The scheme supports static or dynamic CMOS logic elements, and is thus fully applicable to all types of CMOS systems. We will defer the discussion of specific circuit topologies of the latch elements until Chapter Five. For now, in latch based schemes, we can deal only with the abstraction of N-blocks and P-blocks, where "blocks" consist of only their respective latch elements together with combinational logic. We note that the notion of combinational logic elements contained in or between latch elements is equivalent. The previously defined concepts of edge-based systems still hold. A positive edge flip-flop is made from a cascade of a P-latch followed by a N-latch. Similarly, a negative edge flip-flop is made from a cascade of a N-latch followed by a P-latch. .. 0 att 2 negative skew positive skew Figure 3.6: System model for TSPC edge based clocking. TSPC Edge Based Systems The effects of clock skew in a TSPC scheme can be examined with the system shown in Figure 3.6. Skew is defined as the difference in time between clock signals arriving at the receiving flip-flop relative to the transmitting flip-flop. Skew may take on a negative or positive value as shown, depending on the relative magnitudes of the path delays (A, and A2). We note that negative skew has a positive value for A21 and positive skew has a negative value for A21.- We will also define 7, as the setup-time, which is the minimum amount of time that the data must be stable prior to the active edge of the clock. Likewise, we can define 7-h as the hold-time, which is the minimum amount of time that the data must be held after the active edge of the clock. There are also delay times associated with the propagation times for new data through the flip-flops and combinational logic circuits. We will define the propagation delay through the flip-flops (i.e. after the active edge) by 7Q, and the propagation delay through the CL logic by TCL. All of the quantities defined so far may take on maximum and minimum values, which will be denoted with subscripts M and m, respectively. .. (a) I 'TQM +'tCLM -A21 I (b) M Ai21I 'tOM +tCLM Figure 3.7: Timing waveforms for TSPC edge based clocking. As before, we are interested in deriving a set of constraint equations to characterize our system. We wish to be able to run the system at the maximum allowable clock frequency. We now consider the case with clock skew when non-local flip-flops are communicating. The timing diagram for the situation where A2 > A1 (negative skew) is shown in Case (b) of Figure 3.7 The maximum allowable clock period can be expressed as: To __ TQM + TCLM + Ts A21m (3.4) On the other hand, the minimum delays must be such that the newly generated data does not reach the distant flip-flop before its hold time. This requires that: TCLm . TQm Th + A21M (3.5) The maximum allowable clock skew in the system is given by: A21M TCLm + TQm Th (3.6) .. Figure 3.8: System model for TSPC latch based clocking. Typically 7rh is near zero for this class of circuits, so the maximum allowable clock skew is just given by the minimum flip-flop and logic delays. The situation for positive clock skew is shown in Case (c) of Figure 3.7 (-A21). Substituting in a negative value for A21 in Equation 3.4, will yield an increase in the maximum clock period. Thus, positive skew will tend to slow the system down. Negative skew will allow the system to operate faster, but will place constraints on the minimum speed of the logic, since fast CL networks will tend to have shorter minimum delay times. TSPC Latch Based Systems We can obtain a similar set of equations for non-local communication between modules in latch based TSPC systems. Our prototype system is shown in Figure 3.8, and timing waveforms are shown in Figure 3.9. For simplicity we will assume that the delays in the P and N latches are the same. Data stored in the N-latch (when q$ goes low) must propagate through the CL network and arrive at the P-latch before its setup time. This is shown in Figure 3.9, Case (a). For negative skew (Case(b)) we thus obtain the following expression which is very similar to Equation 3.4: .. (a) I TOM +'CLM c) A21 'rQm + CLM -C)A21 4 (C) I. ';OM +rCLM Figure 3.9: Timing waveforms for TSPC latch based clocking. T9L TQM + TCLM + Ts A21m (3.7) We note that there is also a constraint on the high width of the clock in order to give N1 enough time to obtain data. This is given by the following equation where TrQM is the maximum delay of the P latch 70,, QM + T., (3.8) Again we must consider the minimum allowable delay in the system. Data must not race through the P latch and the CL network before the hold time of the N2 latch. Thus, the requirement below must hold TCLm + TQm Th + A21M (3.9) The maximum allowable clock skew in the system is given by: A21M : TCLm + TQm Th (3.10) .. Again, positive skew has the effect of lengthening the clock period. Design of these systems must strike a balance between Equation 3.7 Equation 3.8 and Equation 3.10. It is interesting to consider situations in which Equation 3.10 cannot be satisfied [44]. This is shown in Figure 3.10. Here the skew has exceeded the gate delay. Non- deterministic behavior of the system results because the evaluate phases of N and P blocks overlap. The system can be made to operate correctly, however, by simply clocking against the data flow. This is shown in Figure 3.11. In this way, the evaluation phase of the next block will be completely contained in the data-stable zone of the last block. It is thus possible to exploit the skew, if its direction can be guaranteed. Another solution is to only latch data on the start transitions of the evaluation phases. This can be accomplished by adding a "re- synchronizing" N-block in front of the P block as shown in Figure 3.12. This illustrates the flexibility of the technique, where we can selectively add edge-triggering where needed. Finally we note that in systems where the data flow must be in two directions, a totally edgebased scheme is best. The maximum skew in this case is nearly up to a half clock cycle [44]. 3.2 Synchronous vs. Asynchronous Systems Since the early days of systolic arrays, it has been realized that clock synchronization over a large silicon area would be a limiting factor. In an attempt to solve this problem, Kung [15] proposed his Wavefront Array, in which data would move between processing elements in a self-timed manner, via a handshaking protocol. More recently, Afghahi and Svensson [2] have conducted a study of synchronous and asynchronous clocking schemes, for layout groundrules ranging from 3 to 0.3 tim. The study was based on physical entities in actual processes, rather than on speculative models. The results of their findings have important consequences for .. Data N-Block I P-BOCK N-BlOc CK -bIOCK Clock Ac Ac C1C2 C C4 Ci E L E L-g Output Data 1 + Output Data2 2 c Eval.4~ Corrupted by data 2 data1 a C2 L E L E Figure 3.10: Potential skew hazard with TSPC scheme [44]. Data N-Block P-Block N-Block P-Block C Ac 0, Cs A CClock Ci E L E L 4Agi Output Data 1 Output Data 2 Ac- c Eval. Eval., datal data 2 02 L E L E Figure 3.11: Clocking against the data flow to exploit skew [44]. .. Data --N-Block P-BI- -- N-Block C Clock 02 - C 04 CFE L L FE _L HAg- ~ Output Data 1 -+ Output Data 2 C E EL L E LL g Output Data 3 Output Data 4 H C4 L E L E t Eval. i Eval. -data 1data 2 &data 3 1& data 4 t1 t2 Figure 3.12: Non-local communication problem solution [44]. the clocking scheme used for a particular implementation, since the optimum choice (in terms of overall system speed) is intimately related to the module grain size. Some of their findings are reproduced below, as they are relevant to our problem. Asynchronous timing schemes have been proposed as a solution to clock skew in VLSI systems. It is generally realized that fully asynchronous logic (i.e. no clock) requires too much handshaking overhead for practical use when the number of inputs and outputs is large. Present day interest in asynchronous systems centers on locally synchronous, globally asynchronous schemes, where each module in a system operates with its own clock, but communicates with other modules asynchronously. Since data arriving at a module must be processed synchronously, a synchronizer circuit is necessary for each input. The synchronizer is vulnerable to error, if the input signal changes during the edge of the local sampling clock. If this occurs, the synchronizer goes into what is known as a metastable state (MSS), where the output is undefined for some period of time. A synchronization failure potentially results in a system failure. In order to avoid this, a time interval must be allowed for the synchronizer to resolve the MSS. This time interval is an extra delay which impacts .. the overall data rate. Obviously, reducing the module communication rate decreases the likelihood of synchronization failure, at the penalty of reduced system bandwidth. A more favorable approach, is to develop some probalistic bounds on the resolution time, t, of the synchronizer circuit. In this work, to estimate t, a synchronization failure rate of 1 per year was considered acceptable. Synchronization time was shown to be the limiting factor for asynchronous systems. The study showed that the time complexity of synchronous systems is (O (log R)0-') while for asynchronous systems it is O(logR) where R is the size of the system. This suggests that for fine-grained systems (where skew is acceptable) that synchronous clocking should be used, while for coarser grained applications, asynchronous schemes should be used. The authors also showed that where a pipelined clocking mode is used (i.e. the global clock is segmented into an optimum number of small sections and each section driven by a repeater), that synchronous systems will always outperform asynchronous systems. This is very different from the general belief that asynchronous systems will be the fastest possible implementation in a scaled technology. They also showed that speeds up to approximately 2 Ghz. can be achieved in single phase synchronous CMOS systems for 0.3 jim technology, although the line repeaters must be spaced 1 millimeter apart to achieve this. .. 3.3 Fundamental Manufacturing Limitations The occurrence of defects is inherent to the semiconductor manufacturing process. These defects arise due to many physical mechanisms such as particle contamination, imperfections in insulating oxide layers, mask misalignment, step-coverage problems, warping of the wafer during high temperature steps, etc. The first two cases tend to produce random defects, which affect local regions of a wafer, while the remaining three tend to produce defects which affect many chips (global regions) on a wafer. Global defects are usually not considered in defect-tolerant analysis, since they represent gross disturbances in the manufacturing process, and are generally not a function of chip area. Additionally, various process monitors (test circuits placed at strategic locations of a wafer to determine the quality of each manufacturing step) are used to reject (or correct) wafers which exhibit such characteristics early in the process. Consequently, the overall quality of the manufacturing process is determined primarily by local defects. Integrated circuit yield is intimately related to manufacturing defects. Yield is defined as the ratio of the number of working chips on a wafer to the number of chips fabricated, and is always less than 1. For conventional VLSI systems, yield studies pertain to manufacturing economics with the goal of maximizing profits. In the context of ULSI and WSI systems, however, yield relates to fundamental feasibility. Yield is further divided into two broad categories: functional yield and parametric yield. Functional yield (sometimes called catastrophic yield) is determined based on the criterion of a chip successfully performing its desired logical functions. Parametric yield is determined based on the chip meeting some predefined operating specification, such as minimum speed or power dissipation. For our purposes, we will not consider parametric yield since it is usually highly correlated to global disturbances on a wafer. .. 3.3.1 Defect Size Distribution Early work on defects occurring in the semiconductor manufacturing process modeled random defects as dimensionless points, where any defect occurring in an integrated circuit was assumed to cause a failure. A more modern view of defects [28] models them as extra or missing disks of material in the conducting and nonconducting layers of an IC, which are characterized by varying radius and spatial distribution on a wafer. These may take the form of shorts or opens in conducting layers used for interconnections, oxide pinholes in insulating layers which can cause a short (or leakage) between conductors, junction leakages or shorts to the substrate in diffusions, etc. The model assumes that defect types are independent, in that a defect of a particular type does not interact with, or cause, another defect of a different type. In practice this has been shown to be a good approximation [39, pp.lA9-l7l]. Examples of actual defect types from early prototype fabrication runs of the PE are shown in Figure 3.13. These defects may cause an immediate fault, as in Case A (left), or may not cause a fault as in Case B (right). Case A shows a short between two power busses, which was caused by a particle on the chip (the particle is the dark spot). Case B is a pinhole in the passivation layer (overglass) layer, which does not cause a short circuit since there is no conductor above it. Clearly, then, we must consider the nature of a defect in order to determine if it will manifest itself as a circuit fault. Random defects (sometimes called spot defects) are typically caused by particle contamination (dirt) on the chip or on photolithographic masks during manufacture. As alluded to in the previous paragraph, defects are of many differing types. The size distribution function for defects of type i is given by: fi (R) = kiR 0< R < Xo (3.11) Sk2/(R)Pi X0, < R .. Figure 3.13: SEM photographs of defects from early PE fabrications. where R is the defect radius and XOj and pi are parameters which are extracted from the fabrication line. Each manufacturing step in a process has associated with it its own characteristic set of defect types. The parameter i in Equation 3.11 can thus be taken on a mask-by-mask basis (i.e. consider polysilicon shorts and opens, first-level metal shorts and opens etc.). Smaller defects tend to be more numerous than larger defects, since it is more difficult to filter small particles from the ambient environment and manufacturing chemicals. We see in Figure 3.14 that the defect density peaks at the value for X0i. This corresponds to the resolution limit of the photolithography in the process. The physical reason for the peak is that defects of a smaller radius simply cannot be resolved by the photolithography, and thus manifest themselves with decreasing frequency. Typically, minimum design rules are set well above this value, so that the only size distribution exhibited in practice is fi(R) = k In the (R)P, above expression, both Xoj and pi may have different values for each defect type i. Typically, a value of pi P 3 is assumed [28]. As mentioned previously, not all defects in a semiconductor process cause faults. For example, Walker has suggested that if XOi = 0.5 pm and the minimum .. f(R) Minimum design rule. Defect Radius (R). Figure 3.14: Defect density vs. defect radius line separation is 3 /m (i.e. minimum design rule in Figure 3.14), then only one-inseventy-two defects are potentially fault producing [39, pp.41]. With this in mind, we must then consider the effective defect density, D0i, which is the defect density as seen by the layout. The defect density variation with respect to defect radius, Di(R), is thus obtained by multiplying D0i and fi(R). Hence, D i(R ) (R)P, = fi ((3.12) (R)Pi The relationship between parameters Ki, pi and the effective defect density is as follows. Suppose that two metal lines in an IC have a minimum spacing s, and we are interested in determining the effective defect density for shorts. All defects with radius less than s/2 will not be able to cause a short in the layout, regardless of where they lie. The effective defect density, Di-effective, as seen by the layout is then D t jo K dR= Ki J/2 (R,---- (p- 1)()- .. i 2P-)S (3.13) Similarly, the effective defect density for opens in lines of minimum width w, is given by: Di.ef fective = (pi 1)(2)P:1l (3.14) We have seen in the case of this simple example, a relationship between effective defect density and the layout geometry. There is thus a finite probability of a particular defect producing a circuit fault. This idea is further extended into the concept of critical area, which is the portion of a layout sensitive to a defect of a particular radius. Critical area A(R) can be defined for a defect of radius R, as that area on a die in which the center of a circular defect has to fall for a fault to occur in a circuit. This is illustrated in Figure 3.15, where we consider again, the case of a chip consisting of wires of width w spaced s units apart. The total chip area is A0. As we have seen before, defects of radius less than s12 will not cause shorts, and thus, A(R) = 0 for defects with radius R < s12. If we consider defects with radius R > -s/2, A(R) increases. The critical area increases linearly until it reaches A0. This occurs at defect radius ft0 = (.5 + w12), where a fault occurring anywhere on the chip would cause a short. The increase in critical area is linear for this simple case, but for a complex layout, we can only state that critical area will increase monotonically with defect radius. In practice very large defects seldom occur, and the distribution given in Figure 3.14 can be truncated at some maximum radius. .. Critical Area. R < S/2 S/2. Defects never Some defects Defects always cause short. produce short. cause short. A A(R) Ao S/2 S + W/2 Defect Radius (R) Figure 3.15: Critical area for Parallel Conductors. 3.3.2 Defect Spatial Distribution and Yield Models Poisson Statistics The spatial distribution of defects must also be considered when determining circuit yield. Early yield models assumed that defects followed a Poisson distribution, and were considered as point defects. Poisson processes are modeled by the expression: prob{X = x} = (3.15) The three underlying assumptions of a Poisson spatial process are [16]: (1) that the number of events occurring in one segment of space is independent of the number of events in any nonoverlapping segment; (2) that the mean process rate A must remain constant for the span of space considered; (3) the smaller the segment of space, the less likely it is for more than one event to occur in that segment. For yield purposes, we are concerned with the case where X = 0, as this is the condition .. for zero circuit failures. The mean process rate, A, is number of circuit faults. We can define this per process step by Ai, which is given by Ai = D0iAi, with Doi and Aias defined previously (i.e. the defect density and critical area, respectively). The yield of a particular step in a semiconductor process is thus: The total yield of the chip is then the product of the individual yields, or simply: n = fi Yi (3.17) where there are n total defect types. The formal methodology in Equation 3.16 and equation 3.17 of considering defect mechanisms on a mask-by-mask basis with regard to layout dependent critical areas is best left to CAD tools. For analytical purposes, we usually consider overall average defect densities only. For this reason we will drop the i subscript in subsequent discussions, and define the quantity Do to be the average fatal defect density. Critical areas are usually taken to be the area of the entire chip, or the areas of major subsections of the chip. The yield for Poisson distributed defects, thus becomes: Y =CA =e-DOA (3.18) It is still possible to calculate the individual yields of particular sections of a chip with this simplified view, from the area and defect density of that particular section. Equation 3.17 suggests that areas on an integrated circuit which are the most complex (i.e. require more manufacturing steps), will exhibit the lowest yield. This is the case for logic, which usually makes use of all design layers (typically greater than 12 masks). Conversely, interconnection busses which consist of one layer, say second level metal, with no other layers beneath them, only require a few processing .. steps (three or so). The corresponding yields of such sections will be higher than that of same area logic circuits. The statistical independence of failures modeled by the Poisson distribution permits the total yield of a chip to be computed from the product of its individual component parts. This is the most useful property of Poisson based yield models. Modified Poisson Statistics It is now well known that for large chips, the Poisson model gives pessimistic predictions for yield when compared to actual fabrication line data. This is because of a phenomenon known as defect clustering. Defects tend to cluster between lots of wafers, between wafers in the same lot and across individual wafers [11]. Clustering of defects between lots is perhaps the easiest behavior to appreciate, due to the batch oriented nature of the semiconductor process. Manufacturing parameters will certainly change over a period of weeks or months, due to equipment and operatingenvironment variations. Thus, it is reasonable to expect some statistical differences in the yields of identical chips from different batches. It is interesting to consider the physical mechanisms relating to the other types of defect clustering exhibited. Clustering within wafers is due to minute differences in environmental conditions at different locations on the surface of a wafer. Snapper has suggested that the defect clusters are generated when vibration or other environmental changes (i.e. irregular gas flow or pressure changes) cause a cloud of particles to break loose from the manufacturing equipment [32]. When these clouds land on the surface of a wafer, the resulting defects produced will be clustered. Other very subtle mechanisms also influence the spatial characteristics defect patterns. For example, it was observed for many years in the semiconductor industry that defects tended to be random within the center of a wafer, and correlated towards the perimeter. This caused many researchers to divide wafers into concentric zones, where the yield in each zone was modeled by a Poisson distribution with its own defect density. This .. behavior arose because wafers were carried in plastic boxes called "boats", between process steps [32]. The inside of a "boat" is similar in construction to the inside of a slide projector, where grooves permit wafers to be stacked vertically (in parallel). A boat is open on one side only, and dust particles can only approach the wafers within from this side. Suppose a dust cloud is present near a group of wafers in a boat. Even if the particles are uniformly suspended, they are electrostatically attracted to the nearest edge of the wafers due to the electrostatic potential created when the wafer is slid into the boat. This leads to the observed edge clustering. These defect patterns are less seen today, due to improved wafer handling techniques. We also note that the dust cloud would only affect wafers closest to it in the boat, and may not affect other wafers in the same boat. This explains clustering from wafer-to-wafer in the same lot. The mechanisms previously described, suggest that if there is a defect present at some position on a wafer, that it is highly likely that there is another one near by. This directly violates the spatial independence assumption of a Poisson process, which assumes that a defect at a particular location is not correlated with an adjacent defect. Thus, defect locations on a wafer are statistically dependent, rather than independent. This is why purely Poisson expressions do not accurately model integrated circuit yield. Clustering improves yield since it is better to have defects clumped together affecting fewer chips, than randomly distributed, potentially affecting more chips. The yield formula of Equation 3.18 can be modified to account for defect clustering by assuming that defects are still Poisson distributed, but considering A to be a random variable. The mere fact that A is a random variable suggests defect clustering, regardless of the distribution used [11). This technique was first described in the literature by Murphy [19]. If F(A) is a cumulative distribution function for the average number of faults per chip, then associated with F(A) is the probability density function f (A) given by: .. f(A) =dF(A) fdA= A (3.19) where f(A)dA is the probability of having an average number of faults per chip between A and A + dA. The overall yield thus becomes: Y = e-'f(A)dA (3.20) The function f(A) is known as a compounder or mixing function. Murphy reasoned that a bell shaped Gaussian distribution would be appropriate for f(A), although he could not integrate the resulting expression. He approximated the Gaussian distribution with a triangular distribution. The yield expression he obtained matched his manufacturing data more accurately than a Poisson distribution. Many distributions for f(A) have been suggested in the past, but none have gained more acceptance that that first used by Stapper, the Gamma distribution [32, 30]. The Gamma distribution is given below as: f(A) -r-1 Aale# (3.21) where a and P are parameters. The mean and variance of the Gamma distribution is E(A) = ap3 and V(A) = a/. If Equation 3.21 is substituted into Equation 3.20 and solved, the following expression results: prob(X = x) x!(a)(1 + )+ (3.22) This distribution is known as the Negative Binomial distribution. The average number of faults per chip (the grand average) is normally taken to be A where A E(X) = af, so that/3 = Equation 3.22 can thus be expressed as: prob(X = x) = + (3.23) xw!(a)(1 + -),+3. .. The mean and variance of Equation 3.23 are given by: E (X) A -(.4 V(X) =A(1 + )(.4 we note that the variance of this distribution (U2) is greater than the mean, which is different from the Poisson distribution where the variance equals the mean. The parameter a is determined from data on the distribution of defects, and can be calculated from the expression: ce (3.25) U2- A We see from this equation, that as the variance of the defect distribution approaches the mean, the value of a approaches oo. This corresponds to the Poisson case. Small values of a correspond to increased clustering. We can thus account for the Poisson distribution by choosing an appropriate value of a. This suggests that Negative Binomial statistics are more fundamental to IC manufacturing than Poisson statistics. For all practical purposes, a > 10 adequately models the Poisson case. As before, we are interested in the case where X == 0 for yield and Equation 3.23 becomes: Y = prob(X = 0) 1 ( + c'= (I+ D0Ay (3.26) This expression is known as the Negative Binomial Model (NBM), and has been widely used in the industry to forecast integrated circuit yields. The NBM formally contains an additional gross yield term, YO, which multiplies Equation 3.26. This term models the effects of large scale process disturbances, but as before, is generally not considered for preliminary yield analysis, since it is not a function of chip area. Conservative values for Do are between 1 and 2 fatal defects per cm2 and a < 1. An average defect density of 1 per cm2 has remained the defect standard for some time (although probably now less than 0.5 per cm 2 is more appropriate for .. some fabrication lines). It was recently reported that a facility could be considered "world-class" if it was able to maintain an average fatal defect density of 0.3 per cm 2 (in 1992) [41]. In the past, defect densities have kept commercial IC die areas below 2 1 cm Clustering has been shown to apply to large areas of a wafer, which typically exceeds the area of a chip. This is known as large area clustering. If large area clustering is assumed, it is common in analysis to assume that defects are uniformly distributed within the cluster (i.e. within a chip). We note that this was the assumption implicitly made in Equation 3.26, in that the same value of a was applied to the entire chip. To determine if large area clustering holds for a chip of a particular size, it is necessary to examine particle distributions on actual wafers. This is accomplished by first dividing the wafers into square regions called qaudrats. The number of particles which occur per quadrat is then counted so that a frequency distribution for the number of particles per quadrat can be obtained [29]. The parameters of the Negative Binomial distribution can then be determined from a maximum likelihood estimation technique and checked for goodness of fit with a chi-square test. The quadrat area is varied and the process repeated. The variability in the estimates of a can be obtained. The validation of the large-area clustering assumption is based on the overlap of the standard deviations (u,) of the estimated values of a for increasing quadrat areas. For quadrat areas up to a critical value, the ranges will overlap indicating that a can be considered constant. Beyond this point the ranges will no longer overlap, and the value of alpha will begin to increase. Equation 3.26 will no longer be valid beyond this point. The large area clustering assumption has proven to be valid for well over a decade and a half. Equation 3.26 has been used successfully in the industry for many years and for many products. In our yield analysis, we will assume large area clustering. As chip sizes approach wafer scale dimensions, however, it is very much .. 45 a open question as to what yield model is most appropriate. For very large chips the relationships between clusters is critical, as it impacts the amount and type of redundancy needed for fault-tolerance. What essentially happens for very large chip areas is that there is clustering of clusters. There have been several models proposed [31, 22, 38] to evaluate WSJ or near-WSI yields, but there are no standards as of yet. .. CHAPTER 4 SYSTEM ARCHITECTURE 4.1 Architectural Overview Linear systolic arrays can be used to implement a variety of DSP algorithms such as convolution, FIR filtering, Fourier Transforms and polynomial operations [33]. Linear arrays can also be used to perform linear algebraic operations such as matrix-matrix and matrix-vector multiplication [13]. Linear arrays have reduced input output requirements compared to two-dimensional arrays and vector arrays, which remain constant as more processors are added. This is a key point, as large-area chips are typically I/O constrained. Figure 4.1 depicts the architecture of our proposed system. It consists of sixteen multiply-accumulate processing elements (PEs) connected to form a fault tolerant linear array. Here, a PE which has failed is bypassed completely, and replaced by a spare. There is one spare PE per modulus for both normal and conjugate channels. If more than one PE has failed per modulus, then the system can still operate with a reduced number of PEs. We note that it is possible to obtain full utilization of all good processors in a linear array, which, in general, is not the case in a two-dimensional array. 4.2 Multiply Accumulate PE Architecture Figure 4.2 depicts the architectural details of the GEQRNS processing element, and a die photograph is shown in Figure 5.7. The PE has been optimized to perform complex multiply-accumulate type operations on both in-place or partial result data. Two eight-bit operands to be multiplied, x and y, are the exponents of elements ax,ay E GF(p) 10}. The y operand bus (Y-bus) supplies data to the .. CRT-' R Xr6 PE PE PE PE z (z+z xr-jxi o PE PE CONJUGATE PE PE y X CHANNEL Xz .iIi 2(z z) Xi PE PE NA CNOARP z 21(z+z) Jyr+jyi loHo . A i PE PE CONJUGATE PE PE PEP yr-ji CHANNEL z' 2t(zz) yr-jyi log Hf local storage at each PE in the array, since there are some algorithms for which the Xi -0 CHANNEL EP j 2z flow (whether in the same or opposing directions). An example of this is matrix log PE PE CONJUGATE PE PE P CHANNEL lzyr-jyi log Figure 4.1: System Architecture multiplier directly. The x operand bus (X-bus) supplies data to the multiplier and to the input of the data-storage shift register (DSSR). It is desirable to incorporate local storage at each PE in the array, since there are some algorithms for which the data dependency does not permit operands to arrive at each PE via a purely linear flow (whether in the same or opposing directions). An example of this is matrix multiplication, which typically employs a 2-D, or vector array. For signal processing, however, matrix multiplication is normally restricted to the case where at least one matrix is pre-known (usually filter coefficients), or even further to matrix-vector multiplication (i.e. signal vector). We can thus pre-store the columns of the known .. matrix for matrix multiplication (or block multiplication, for large matrices), and linearly propagate the rows of the input matrix or signal along the array. This results in greatly reduced 1/O requirements. In this implementation, the DSSR can be used to store up to sixteen operands which are known a priori. A shift register was chosen over a SRAM since it requires less area than a sixteen by eight-bit SRAM (i.e. no decoders or read/write circuitry) and requires minimal control logic. As it stands, the DSSR is approximately 60% of the size of the multiply- accumulate portion of the PE, which illustrates how area-expensive programmable storage is. If more storage is desired for a particular application, then at some point a SRAM will become more area-efficient, even with the associated increase in overhead. Once data has been loaded into the DSSR, it can be circulated continuously via an internal feedback path. The X-Bus can be freed for other global data-move operations when local prestored data is used. Data shifting in the DSSR can also be halted (which is a significant feature since we are using dynamic latches). This will greatly simplify array data-flow timing, as processors further along the linear chain can wait for operands /partial results from previous processors, with their prestored internal operands "lined up in place" and ready for processing. Just prior to multiplication, it is necessary to check for a zero operand, since zero must be handled as an exception in GEQRNS. An unused binary code word is chosen as the GEQRNS zero (i.e. some value between pi and 28 1). In this case, 255D was chosen, since zero detection will simply be the logical AND of all data bits. If a zero is detected for either operand, a flag is raised, which will set the input of the exponentiation table's pipeline register to zero after the next clock cycle. The output of the modular adder provides the address input to the exponentiation table. We note that we could have used a standard binary adder here, and performed the modular reduction in the ROM. It is much more area efficient, however, to perform .. Figure 4.2: Processor Architecture the modular reduction first, since the size of the ROM is halved. This is evident from a comparison of the relative size of the modular adders and ROM in Figure 5.7. A doubling in ROM area would represent a substantial increase in PE area, whereas our modular adders are only about 50% larger than a similar word-width binary adder. The ROM table has the value of a"'fX+Y i-i programmed at each corresponding address location. The QRNS product is thus obtained at the output of the ROM. The computed product is then fed to one input of another modulo adder, which reduces the computed sum mod(pi). If the second input of this modulo adder is connected in feedback to its output, an accumulator is formed. This mode is used for algorithms requiring results to be computed in place. For algorithms requiring .. partial results to flow from PE to PE, the second input. of the mod(pi) adder is connected to an adjacent processor. A dedicated bi-directional systolic output bus is used to transfer computed QRNS results (in place or partial) to the next adjacent processor in the array, in the natural ordering predicated by the algorithm being implemented. 4.2.1 Modulo-P Adders Figure 4.3 depicts the basic construction of the modulo adders used for the multiplier and accumulator portions of the PE. This is a standard biased-addition scheme [9], where an offset of value 2n~ p, is added to a 2n bit adder to make a mod(pi) adder (i.e. any n-bit binary adder is intrinsically mod(2')). Operation of this scheme requires that the input words be E f{O. (pi 1)} which is accomplished during input conversion. The magnitude of pi is also required to be less than 2' 1. Two binary adders are cascaded such that the output of the first adder is input to the second adder which has an appropriate offset added. Here, eight bit ripple carry adders are used, and the offset added to the second adder is the value (28 pi). The correct mod(p2) sum is selected from the outputs of either the first or second adder via a multiplexer controlled by the logical OR of the carry bits of the first and second adders. In most implementations, standard cell adder modules are used, and the offset programmed by hardwiring inputs to low or high as needed. This is somewhat wasteful as the offset is known a priori, and thus half of the logic of the second adder need not be included. In this implementation, the offset has been bit programmed by only implementing the logic corresponding to an added zero or one at each bit position of the second adder. Basic mod(p) adder primitive cells are then constructed from a FA and the logic corresponding to offset-zero and offset-one (see Figure 4.4). The transmission gate adder presented in [42, pp.317-320] was used to implement the FA circuit, since pass-gate implementations were found to be faster than standard .. Iott4- M .oI- o x V Figure 4.3: Standard Modulo P Architecture Table 4.1: Full Adder truth table. FA Truth Table Inputs Outputs Ci Ai Bi Sj+j Cj+j 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 CMOS realizations. These blocks can then be used to construct a mod(p) adder of arbitrary value. Since the datapath width is only eight bits, a ripple carry scheme can be used here without speed penalty. For larger wordwidths, such as needed in the CRT, a carry select scheme can be used with these same small width ripple carry modulo adders (Figure 4.5). Here, the larger dynamic range is partitioned into m, k bit sections. The offset value of corresponding to 2mk M, where M is the desired modulus is programmed as before by selecting and arranging the appropriate primitive cells. We can think of these zero and one primitives as having two input and output carry bits, and a multiplexer select input. The four possibilities for the input carries can be preprogrammed and selected by the output carries from the previous k bit section. The total modulo add time is thus: .. C24, ONE I PRIMITIVE 01i Q uN Cl 02 uHli Figure 4.4: Modulo-P Adder Building Block Primitives. T,,odM = T kiple + (m 1)Tmux4 + TOR + Tmux2 + 7mux4 (4.1) where, Tmux4, TOR, Tmux2 are the propagation delays of the four-input carry multiplexer, the final OR-gate and the two-input multiplexer (in the primitive cells), respectively. For typical technologies, these times will be all < 1 ns (including wiring delays over the distances involved), and the delay for a single primitive cell is ins. The area required for an eight bit modulo section is 708 A x 140 A (283 ftm x 56 /im in 0.8 jzm technology). Thus, replicating the k bit sections 4(m 1) times will consume minimal area. 4.3 Forward Mapping: Integer to GEQRNS Equation 2.5 and Equation 2.6 describe mappings necessary to convert input data to and from QRNS, respectively. These equations are implemented as shown in Figure 4.6 and Figure 4.7, respectively. For the forward mapping, it is necessary to reduce the real and imaginary components of the input eight-bit data streams C161 A .. Y(mk-I).(m-I)k -S> C1CO C1CO CICO C1CO o0 0 1 10 1 1 *+t C1 CO M C1 CO M C1 CO M C1 CO X(2k-1),k Y(2k-1),k -+ -C01 CO C01 CO C1 CO 01C 00 01 10 1 1 C1 CO M X(k-1),O Y(k-1),O -) C1 CO 00 Figure 4.5: Carry Select Modulo Adder. mod(pi), since the correct operation of the mod(pi) adders requires that the input operands be E {0. pi -1}. This is accomplished by two 256 by eight-bit ROMs. For the imaginary part of the input words, multiplication by +3 and modular reduction is also accomplished in the same ROM. We note that +3 is used for the normal channel and that -3 is used for the conjugate channel. The ROM outputs are then input to a modular adder to complete the QRNS mapping. The QRNS operand is then converted to GEQRNS via a final logarithm table, which has the zero encoding present address zero. Again, it is more area-efficient to reduce the sum modulo Pi first, rather than perform this operation in the ROM, as the size of the ROM would .. Figure 4.6: Forward-mapping (0) conversion module with GEQRNS log table approximately double. We note that four of the modules shown in Figure 4.6 are needed per modulus (see Figure 4.1). 4.4 Inverse Mapping: QRNS to Residue The architecture of the inverse QI{NS mapping is shown in Figure 4.7. Input data arrives from the output of the last PE in the array. Equation 2.6 requires that we perform a z z* operation for the imaginary part of the output. This is accomplished by looking-up the modular complement of z* (i.e. -z + pi) before adding to the z term. This keeps the inputs to the modular adder in the desired range for correct operation (i.e. E {0o. pi I}). The result of the modular addition is used to look-up the value of 2-131 times the input to the ROM, with a final mod pi reduction. The operation of the real part of the inverse mapping is likewise analogous, except that normal and conjugate operands can be added modulo pi directly, and that 2-1 is looked-up times the input to the ROM, with a final mod pi reduction. 4.5 Chinese Remainder Theorem The architecture of the CRT is shown in Figure 4.8. Here the real and imaginary outputs of each of the modules in Figure 4.7 are grouped and fed to real and imaginary CRT modules (see Figure 4.1). This is denoted by the xi inputs in .. regfsters" P -g-A ., to -e <2-1(Z+Z*)>pi (REAL) Z -, / 8 ROM (265 x 8) d+ to <2 (IMAG.) ROM (265 x 8) _A+<_Z,>pi (complement) Figure 4.7: Inverse-mapping (0-1) conversion module. Figure 4.8, and likewise in Equation 2.3. We note that the xi terms are the only "unknowns" in the CRT, as rihi and rhy' are pre-known constants. We can thus use xi to look-up the corresponding expansion terms in the CRT, and perform the desired mod(M) reduction at the same time in four ROMs. We could have used 256 by 31-bit ROMs here (the dynamic range is 30.9 bits), but they would be significantly slower than their narrower counterparts due to increased internal capacitive loads (i.e. higher word-line capacitance). It is imperative to keep the delays in the CRT the same as those in the rest of the system, so as not to degrade overall performance. The final modular summation is accomplished by mod(M) adder tree of carry-select modular adders of the type described in Figure 4.5. Thus, all of the input and output conversion hardware can be implemented with the same basic ROM and modular adder cells developed for the PE. The computational throughput is thus the same for all elements in the system. .. A A -1 -ROM (265x7) ROM ROM 31 (265x) u8) ROM (265x8) < <(M2 2)'*X2>P2>M 31 8 -ROM 3 (265x7) ROM .(265x8) " (-x) ROM 31 (265x8) ' ROM (265x8) <>M 8 (A 31 ROM out < <( O4) x4P>M (265x8) ROM (265x8) 8ROM (265x8) (265x8) ' Figure 4.8: CRT block diagram. .. CHAPTER 5 VLSI IMPLEMENTATION OF PROCESSING ELEMENT 5.1 True Single Phase Clocking Scheme A TSPC edge-based clocking scheme was selected for the implementation of the processing element. Edge-based clocking was selected over a latch based scheme for several reasons, the most significant of which is the requirement that the data should be able to flow in two directions. Bi-directional data flow offers the most flexibility from an algori thm- mapping standpoint, which is critical for the linear array. Another reason for the use of an edge based scheme is that the circuitry of the PE is very heterogeneous from a VLSI perspective. The modular adders are combinational circuits and the ROM is a CMOS domino-logic circuit. Furthermore, the accumulator portion of the PE is either a state machine (i.e. new data gets added to the running sum which is fed-back), or a combinational circuit if partial sums arrive from other adjacent PEs. We thus have a combinational multiplier (mod-P adder) which drives a CMOS domino ROM which in-turn drives a state machine or another CL block. Also, the data storage shift register introduces its own set of timing requirements as we will see subsequently. The clocking scheme of the PE was the dominant design issue, which required extreme care to implement successfully. The timing requirements ultimately determined the types of circuits selected. All of the synchronization issues introduced in Chapter Three came into play for the design. The move from NPTC to TSPC clocking was the most significant feature which distinguished this version of the PE from its earlier predecessors. The associated increase in performance was greater than a factor of two. .. v i . MO M2 M SXP SYP DD HM5 (D -cJM3 nD SXN SYN 4M4 M7 M Figure 5.1: TSPC Pipeline Register 5.1.1 Pipeline Registers The split-output latch circuit [44] shown in Figure 5.1 was used to implement all pipeline registers and DSSR cells in this chip. This particular cell exhibits minimal clock loading in that only two transistor gate loads are seen per stage. This is ultimately why this particular variant was selected over others in its family, which have four gate loads per stage [44]. This fact is very important since there are many gates connected to the clock line in a highly pipelined architecture such as this. The DSSR contributes the highest portion of capacitance per PE to the clock line, so halving the clock load per latch is an excellent tradeoff. The split-output latch is more difficult to implement than its counterparts, and must be designed very carefully. This will become apparent shortly. The pipeline registers (flip-flops) are of the negative edge triggered kind. Negative-edge triggering was chosen because it was most compatible with the timing requirements of the ROM. A single large inverter can also be used as a clock buffer, since most external system level clocks are positive-edge triggered. .. 5.1.2 Data Storage Shift Register The data storage shift register imposes the strictest timing constraints on the clock signal. A shift register can be built with the un-buffered (inverting) form of the split-output latch, if there are an even number of stages. This results in a minimal transistor-count implementation. As we are interested in building a shift register which is sixteen levels deep, then the use of this configuration is a possible choice here. There are implicit subtleties, however, in such a scheme. This is because a potential fast-transition race condition exists for the case where the output of a register directly drives the input of another register. If the clock-edge transition time is slower than (or even on the order of) the transition time of the register output, then timing errors may result. This is demonstrated in Figure 5.2 from SPICE simulations on the extracted layout of the un-buffered shift-register cell. The simulation represents the output for a cascade of two cells. The first cell in the chain is driven from a "stable" data source (bitO), which changes well before the negative-edge of the clock. The output of the first cell (bit1), drives the second cell whose output is shown in the third curve (bit2). The clock-edge transition time is swept from 1.75 ns to 3.75 ns, in 0.25 ns increments. At about an input clock-edge transition time of 2.75 ns, a gradual negative slope can be observed at the output of the second latch, during the high portion of the clock. The exact failure mechanism is is complex, and can best be understood by an examination of the fourth and fifth curves. A fast low-to-high transition will cause transistor M7 to turn-off too early (i.e. before M3 fully turns off). This may cause excess charge from node SYP to be deposited on node SYN (which should have been fully discharged), thereby causing its potential to rise. If this potential exceeds the threshold voltage of M6 then an unwanted discharging of the output node will result. The potential of node SYN is augmented by clock-feedthrough when the clock transitions from low-to-high, which exacerbates the problem. We note that this .. behavior is a function of the clock edge-rate only, and cannot be solved by slowing the system down. The clock-edge rate will thus have to be kept relatively fast for the unbuffered case. The total parasitic capacitance on node SYN should be kept as large as is practical, so that the magnitude of the "hop" induced by clock-feedthrough is minimized. For this reason, oversized diffusion islands were used on the drains of transistors M7 and M3 (see Appendix A). The situation can be ameliorated, however, by introducing some delay between latches. The most obvious solution is to use "weak" inverters to buffer the outputs. This was used here, but was taken a step further which yielded a double gain for the tradeoff in area. A feedback path was added to the DSSR cells, which permits the data contained in the DSSR to be "frozen" (note the latches are dynamic). From a data-flow standpoint this is desirable, as we can stop the shift registers in adjacent processors with operands in the correct "place". This means that we do not have to zero-pad the stored data for alignment in the shift register. The DSSR is costly enough in area, and including extra cells just so that they can be zero-padded is an unacceptable waste. The tradeoff, of-course, is more transistors per cell (and another control signal). However, this was deemed viable during the design phase, as the area of the DSSR is still smaller than an SRAM based implementation. The final shift register cell implementation is shown in Figure 5.3. The maximum transition time of the clock is now approximately 4.25 ns. 5.2 Exponentiation ROM Semiconductor memories provide the key computational elements of most RNS systems and usually determine maximum obtainable operating speed. A ROM was chosen over a RAM for the Exponentiation table of the PE, since a ROM of a particular size is a factor of four to six times smaller than a RAM of the same size. There is also no need for programmability here, as the moduli are fixed. A block diagram .. 61 FAILURE MODE IN SHIFT REGISTER FOR EARLY LOW-TO-HIGH INPUT SRP2A.TRO V L 9.0 I.- PHI 0 1 L N 2.0 . B. . . ITO T LO -J ., I m 0. : , SRP2A.TRO V L 9.0 BITt 0 1 L N 2.0. T i i , 0 . . V . . . . . i S 0. I .i. iii i ii ii[i. . . . . .5. SRP2A.TRO V L BIT2SN 0I A L N 2.50 T 0 . i : 1 r2 SRP2A.TRO V L .0 X2.SXN 0 1 L N 2.0 ? X2.SXP T .-O IRP2A.TRO 0 I: L.L. .J .J. . .J. I.l. .L 0T 25.0N 50.0N 75.0N 100.ON 0. TIME (LIN) 125.ON Figure 5.2: SPICE simulation of fast transition path for shift register .. D D D STT ST Figure 5.3: TSPC Shift Register Cell with Storage of the logical organization of the Exponentiation ROM is shown in Figure 5.4, and the key circuit elements are shown in Figure 5.5. The ROM is masked programmed based on the modulus choice, by including a contact to connect a programming (discharge) transistor to the bit line. Connecting a programming transistor in the ROM array, results in a logical zero at the output of the ROM, for the accessed bit location. There are eight parallel bit locations accessed per address, thereby producing a byte of data per address input. There are a total of 256-bytes stored in the ROM. Since the programming transistors are of minimal width, they will limit operational speed due to the discharge delay of the bit-line. The ROM logic is partitioned so that the bit line height can be kept relatively short (which results in reduced parasitic capacitance). Differential sensing techniques were also used, so that a logical zero could be determined well before the bit line has fully discharged. Figure 5.5 reveals that the ROM is precharged during the low clock level, which is just after new data has arrived at the address inputs. The data is stable during the evaluation (high clock) stage, when the word-line decoders become active. .. 1-2 BLOCK DEC. SENSE AMPLIFIERS 2:4l COLMN DEC: 12 BLOCK DEC. SENSE AMPLIFIERS 2:4 COLUIMN DEC 5:32 128 X 8 bit WORD 128 X 8 bfit STORAGE LINE STORAGE ARRAY DECODER ARRAY Figure 5.4: Floorplan of Exponentiation ROM. SENSE AMPLIFIER BLOCK I I I I DECODER BLy+3 BLy+2 BLy+I BLy < c4~ Figure 5.5: Key ROM Circuit Elements. WLX Vdd .. A precharge PMOS transistor which is controlled by the clock and a "weak" PMOS device connected in a positive feedback configuration have been included at the sense amplifier bit-line-voltage-input. The precharge transistor was included so that this point may quickly charge to Vdd. If there were no PMOS devices here, then the potential at this node would slowly charge to approximately 4V through two series NMOS devices of the column decoder (since at one-of-four paths is always active). The PMOS device in the feedback loop also helps with precharging once the sense amplifier has "tripped", but was primarily included to maintain a high potential at this node (for a programmed high), in case of any charge loss caused by leakage paths. The low cycle duration of the clock can thus be shortened during precharge, so that more time can be given to the high cycle (where bit-line pull down occurs) without adding to the total clock period. This is consistent with the notion of "trading delay", introduced in Chapter Three. Figure 5.6 shows a SPICE simulation of the operation of the ROM, for the case of a programmed low. This represents the worst case delay, as a logical high value is obtained by default from the action of precharging the bit-line. The simulation was done on an extracted layout of a bit-line programmed with all zeros, as this is the worst case for parasitic diffusion capacitance. A sense amplifier and the full word-line decoder were also included, and the parasitic (gate) capacitance of the maximum possible number of programming transistors per word-line (64) was also taken into account. The first graph shows the clock input and address line zero, which becomes active shortly after the clock goes low. We note that even though a valid address is present, all word-lines are low due to the action of dynamic NAND gate and inverter combination of the word-line decoders (see Figure 5.5). The output of the sense amplifier is shown in the second graph. This goes high as expected during precharge. The potential of the sense amplifier input as well as the bit line is shown in the third graph. These two points charge at essentially the same rate. .. The potential of word-line-one is shown in the fourth graph, which becomes active after some delay during the evaluate phase of the clock. When the clock becomes high, the bit line begins to fall. We note the input of the sense amp begins to transition slightly after the bit line, due to the action of the "weak" PMOS device. This point soon "catches-up" to the bit line once the sense amp has had time to turn the feedback device off. The bit-line potential falls as expected. The rise time of the bit-line is approximately 2.4 ns during precharge and the fall time is approximately 3.0 us during discharge. The propagation delay taken from the mid-point of the rising clock edge to the mid-point of the sense-amp output is approximately 5.6 us. The simulation was conducted with clock rise and fall times of 2 ns, and with a clock period of 15 us. The simulation output suggests that the cycle time could be possibly shortened, however, to approximately 12 us. 5.3 Electronic Reconfiguration Switches In an effort to improve the survivability and manufacturability of large linear chains of PEs, electronic reconfiguration switches were incorporated into the PE architecture. These elements are the outermost transmission gates in Figure 4.2. The layout ground rule used for these elements was less aggressive than that of other circuits in the chip, since it is desirable that a very high switch yield be exhibited. For example, a six-lambda minimum metal thickness and spacing was used (minimum is three-lambda), in order to reduce the likelihood of shorts and opens, and double contacts and vias were used between all layers. One author has suggested that extra or missing material defects greater than two line spacings or widths, respectively, are considered so rare that they almost never occur in practice [10]. Indeed, it has been verified consistently for several years, that the defect frequency falls-off with the cube of the radius [28], thus small changes in line spacings or widths can greatly impact survivability. Bypass interconnection lines were also run in metal two, with minimal .. TIMING WAVEFORMS FOR ROM (PROGRAMMED ZERO) 2 . . . O 1. . I. . r L 0 **-BU L !/ A I 2. ROM2.TRO L I.0- :OUT I BLO N 2.0 593 .921U ROM2.TRO L A I N 2.0 .10 : i :. : :. . . 0 i' . .JJ . I . . -ROM2.TRO . . . . . . .; ,\ . . . . . .U. . . V OI N I BLO 2.0 5.ON 10.ON 15.ON 20.ON 25.0N 0. TIME ELIN) 25.ON Figure 5.6: SPICE simulation of ROM operation. .. logic placed underneath these channels, in order to reduce the probability of interlayer shorts due to oxide pinholes. In order to truly determine the fault vulnerability of the reconfiguration switches, it is necessary to have detailed knowledge of the process defect statistics. For example, it has been suggested [40] that due to the use of positive photoresist in most high resolution processes [3], that extra material defects are a factor of ten times more likely than missing material defects. This means that wider spaced, thinner interconnection wires would actually have a higher yield than thicker, closer spaced wires in this process. Thus, the accurate modeling of the interconnection yield within the context of the process defect characteristics crucial. There are advantages and disadvantages to electronic switches verses physical restructuring techniques [36]. Physical switches, such as laser programmed links, have lower on state resistance compared to electronic switches, and typically exhibit smaller area since no other logic is needed (i.e. latches) to store configuration information. The main drawback with physical switches is that they must be programmed at the time of manufacture, and thus, cannot correct for faults which occur in the field. Conversely, electronic switches can correct for run-time failures and can also compensate for manufacturing defects. In a fully pipeline architecture such as this, the added switch propagation delay is contained in the pipeline delay, and, therefore, is less of a design issue. Only one configuration latch is needed for all reconfiguration switches in the PE, since all busses are switched out simultaneously. It is important to note that the operation of the reconfiguration circuitry, only depends on twenty-four transistors (i.e., 12 transmission gates (Figure 4.2)), and twenty-four wire segments per PE. 5.4 PE Performance The processing element shown in Figure 5.7 was fabricated in a 1.5 ym CMOS process (ORBIT Semiconductor [21]) and occupies a total die area of 2.4 mrn x 2.4 .. mm. The chip represents a full-custom design with the exception of the 1/0 pad drivers which were supplied by MOSIS [37]. Data was fed to the chip from an eightbit binary counter circuit. Because of the 40 pin limitation on the package, it was not possible to bring out all 24 input signals (8 x 3) in addition to the power, ground, control inputs and the 8 outputs needed. Instead, both multiplier inputs were brought out and the external accumulator input was fed internally from one of the multiplier inputs (the Y input). It is thus not possible to test the multiplier and accumulator portions of the chip independently. The tests consisted of holding the Y-multiplier input constant, while incrementing the X-input. In this way the contents of the exponentiation ROM could be examined. The Y-input was held at a value of zero (decimal zero), which gets added to the output of the ROM by the accumulator. Both the input signals and output data were simultaneously examined with a Hewlett-Packard 16500A logic analyzer, which sampled data at each rising clock transition. The chip clock is produced internally from a buffered and inverted version of the external clock, so a positive-edge externally is a negative-edge internally (of-course phases are flipped also). Recall that the chip is negative-edge triggered internally. It was possible to verify the programmed contents of the ROM (see Appendix B). It should be pointed out that this is the strictest test, that could be conducted, as the ROM could fail due to insufficient precharge or discharge time. Also, since the ROM is addressed by a modulo adder, the test verifies that the mod adder can sustain the data rate. The observed data was compared to a mask of the programmed data by the logic analyzer, which was setup to stop if there was any difference between observed and mask data. This test was run for approximately three weeks. The chip was run at a clock frequency of 40 MHz. during this test without the logic analyzer stopping. The test was discontinued after three weeks. A bit-line compare voltage of approximately 4.6V was needed to obtain this data rate. This suggests that the bit-lines were not fully discharging at this .. Figure 5.7: Die photograph of processor. .. Figure 5.S: Oscilloscope photo of clock signal and output bit zero. speed, which further supports the use of differential sensing techniques for flexibility. The second test involved holding the X-input at zero (which we recall is is coded as 255D), while sequencing the the (shared) Y-input. In this way, the operation of the modular accumulator could be verified. A photograph of the clock signal and the least significant data output line is shown in Figure .5.8. The least significant data bit is -statistically" the output that changes the most, and timing problems will usually be seen here first. Figure 5.S shows that the output data transitions some several nanoseconds after the rising edge of the clock. This is observed because of the sum total of I/O delay times. That is. the total delay before the new data can be presented externally, relative to our fixed reference of the external rising edge, is the input buffer time 71,B plus the output buffer time TOB. There are also other delays such as latch propagation delay times as well, in addition to the clock internal buffer delay. The output seems to lag the rising clock-edge by a time on the order of 10 to 11ns. In order to oain a better estimate of the I/O delay times involved, a simple pass-through test was implemented. This is just an input buffer which directly drives .. INPUT OUTPUT 'CB+ rOB IOUTPUTL INPUT F Figure 5.9: Pass-through test. an output buffer (the output buffers are inverting). This is depicted in Figure 5.9 and the actual output is shown in Figure 5.10. The time between the peak of the input and the low of the output is approximately 5-6 ns. It is interesting to compare the observed delay time with the simulated delay time. The output of a SPICE simulation of the pass-through test is shown in Figure 5.11. For the simulation, a load capacitance of 10 picofarads was used to model the capacitance seen at the output of the chip. It was not possible to actually measure this capacitance, but 10 PF is a reasonable estimate (the scope probe is 7 pF). The simulated delay time for a high-in to a low-out is approximately 3.1 nS and for a low-in to high-out, approximately 4.4 nS. This suggests that the measured vs. simulated delays are on the order of 25-40 % higher. Although this cannot be generalized for the rest of the circuits in the chip, it still provides a crude measure of the variability between actual and simulated performance. .. 72 Figure 5.10: Oscilloscope photo of pass-through test output. SPICE SIMULATION OF PASS-THRDUGU TEST S .O L T .T O IN L T . L I 5 0 DELAY-T-TRO OUT . I 1 .0 . . . . . . . 6. N 0. N 15-ON 20 ON 2 ON TIME ILINI 2S ON Figure 5.11: SPICE simulation of pass-through test. .. 5.5 Early Versions of the Processing element 5.5.1 Version One Figure 5.12 details the architecture of the first version of the PE and a die photograph is shown in Figure 5.13. This chip did not use the modular adder structure described earlier, but rather, computed the modular reduction of output operands in ROMs. The two input operands were "multiplied" in a standard binary adder and fed to the exponentiation ROM where they were reduced modulo p- I and the generator, a, raised to the corresponding power (all done as a single lookup). A modular adder was made for the accumulator portion of the PE from a standard adder and a final ROM table to perform the modulo p reduction. The PE could only be programmed for seven-bit moduli since, as described in Section 4.2, the result of performing the modular reduction in a ROM table after an addition, is an effective doubling of the ROM size (i.e. the sum of two k-bit numbers is a k + 1 bit number). The modular adders produced from a binary adder and a look-up table were two pipeline stages deep (see Figure 5.12). For the accumulator there is an implicit subtlety here. If this structure is used, then odd and even indexed terms in a summation will be accumulated independently. That is, two partial sums will be formed, one for the even terms and one for the odd terms, which cannot directly be summed. This requires a final summation externally to the PE to complete the accumulation. This fact forced the development of cells that would support the implementation of a single-cycle modulo adder, without compromising speed. At the time, the only cells available [20] to build the modular adders were four-bit carry-lookahead adders which would not have operated fast enough to support an un-pipelined cascade. The bit-programmed modular adder cells described earlier, thus significantly impacted the design. The first version of the PE also employed non-overlapping pseudo two phase clocking. A photograph of the clocking waveforms is shown in Figure 5.14. In this .. figure, the vertical axis is 10x, the lower trace is 01 and the upper trace is 02. The pipeline registers were composed of a cascade of two transparent latches (just as described in Chapter Three). Data was input to the first latch during the high phase of 01 and was presented to the logic circuits on the rising edge of 02. The ROM was precharged during the high phase of 02 (i.e. PMOS precharge devices were controlled by 1-2) and was evaluated on the low phase of 02. Output data was then latched on the falling edge of 01. It was actually necessary to make a third clock signal to avoid a race condition in the ROM word-line decoder, by delaying 42 slightly. The early ROM employed a dynamic NOR-gate decoding structure rather than the dynamic NAND-gate used in the final version. The NOR gate was gated (ANDed) with the delayed version of 72, so that the word-lines would be low during precharge. The gating clock signal had to be delayed so that a momentary glitch would not occur on the word-lines just after precharge, before all but one NOR gates discharged. This complication was eliminated with the NAND structure (although the dynamic -NOR is intrinsically a faster gate). The first PE was fabricated in a 2 /m CMOS technology [37] and was shown to operate at a clock frequency of 16 MHz. 5.5.2 Version Two The second revision of the PE, was the first to use the modular adder scheme in its accumulator. Its architecture is given in Figure 5.15, and a die photograph is shown in Figure 5.16. The removal of the second ROM resulted in a significant decrease in area. A modulo adder was not used as the "multiplier" portion of the chip, due to vertical space limitations in the MOSIS Tiny-Chip [37] pad-frame. This PE thus also used seven-bit moduli. The second chip was also a NPTC macnine, which employed the same ROM and pipeline latches as its predecessor. This chip was also fabricated in the MOSIS 2 lim CMOS technology and was shown to operate at 16 MHz. .. Figure 5.12: Processor architecture of first version. .. Figure 3.13: Die photograph of first version of PE. .. Figure 5.14: Oscilloscope photo of non-overlapping clocks for first chip. EXTERNAL ZERO DETECT Figure 5.15: Processor architecture of second version. .. ~Ij ~ ~3 ~. i:~~ ~' Figure 5.16: Die photograph of second version of PE. L3,712 .. CHAPTER 6 YIELD ENHANCEMENT AND FAULT TOLERANCE 6.1 Yield Enhancement via Reconfiguration It is evident from Chapter Three, that large area integrated circuits must have some measure of fault tolerance. For highly structured architectures, which have identical replicated cells, it is sometimes possible to include redundant modules of which m out of n must function in order for the chip to be considered usable. In the past, many authors have considered the reconfiguration yield to be unity and that all failures could be corrected for by the reconfiguration scheme. Assuming that the reconfiguration yield is 1 is increasingly being shown to be a bad assumption, particularly if the system area is large. What is really assumed, is that m out of n cells are free from defects which affect reconfigurable nets and that all n cells are free from defects that affect non-reconfigurable nets. Actually, we must also consider defects that affect global signals such as power and ground, as they cannot directly be reconfigured for. This requires a very detailed consideration (layout extraction) of susceptible areas on a net by net basis, however, which is more suitable for CAD tools. For our purposes we can consider our FE area as being composed of two components: APE =-,: APE-. + APE-n (6.1) where APE-n is the area of the bypass and interconnection busses and APE-T is the area of the remaining computational-logic portions of the PE. As mentioned previously, the layout ground rule used for the bypass elements was less aggressive than that of other circuits in the chip. Since the complexity .. of these elements is greatly reduced over that of other portions of the circuit, an argument could be made that the effective fatal defect density of the switches is reduced over that of the rest of the circuit (i.e. fewer applicable defect mechanisms). However, in order to truly quantitatively determine the relative fault vulnerability of reconfiguration switches, it is necessary to have detailed knowledge of mask-by-mask process defect statistics. Since we do not have such statistics, we will not speculate, and will consider all areas of the PEs at the same defect density. Our estimates will thus be slightly conservative. Since we are assuming large area clustering, the faults in adjacent modules are dependent (uniformly distributed), and we cannot use simple Binomial expressions for determining if M out of N modules are functioning (this assumes faults are independently distributed). Systolic arrays in which R spare PEs have been added, where at least MN R out of N must function, have been proposed before. The independent, parallel nature of RNS channels, however, provides further unique opportunities for enhancing fault and defect tolerance of systolic arrays. For our proposed four modulus system, each processing node in the array is effectively broken up (algorithmically and physically) into four smaller PEs via the RNS mapping. This would suggest that the inclusion of an extra PE per modulus would permit up to four faults to be tolerated per channel, rather than just one, when compared to a similar system (i.e. in dynamic range and arithmetic functionality), in which the computations are carried out over one physically contiguous PE. The requirement, here, is that no more than one fault occurs per modulus. If there is more than one fault per modulus, then the number of usable PEs will be less than M. Koren and Stapper [11] have presented a yield model for chips with redundancy consisting of multiple module types, in which the failures in adjacent modules are dependent. We can consider our proposed linear array chip as fitting this category, where there are eight different module types (two x four moduli), where M out .. of N PEs must function, and there are R spares for each module type. The following expression describes the total yield: NI N2 N8 Ni-MI N2-M2 Ns-Ms Y= E E E E E .E Ml=N,-RM M2=N2-R2 MM=N8-R8 kl=O k2=O ks=O (l)k k2 k8 ( N1 1- M Ns N- -M8s x I1+ ((Mr + ki)At +. + (Ms + ks),s + ACK) x CM1 ,M2 ,M,MM s,M6,M ,M8 (6.2) where the Ai terms represent the number of faults in each PE and ACK is the number of faults in the "chip-kill" area. The "coverage factor" term in the above expression, CM,m2,M3,M4,M5s,m,M7,M = 1, if the chip is acceptable with M and M2 Ms fault-free modules of types 1. 8. Otherwise, CM,M2,M3,M,,Ms,M6,M,7M8 = 0. This term provides the means of counting those terms in Equation 6.2 which are fixable. We note that all combinations are fixable since the moduli are physically discrete and the system is operable if there are at least M PEs surviving in each modulus. The number of faults in each module Ai is related to the number of faults in the total chip, A, by the following: Ao= (6.3) Atotal For our proposed system, the corresponding parameters are: N = 17 R; = 1 Ai = APEr .. ACK = AIopads + AotCon + ACRT + AICo,,. + 8 x 17 x APEn (6.4) and CM,M,M3,M4,MS,MG,M7,M, = 1 for Mi E {16,17}, since all of these combinations are fixable. We note that the chip kill area contains the area of all input and output conversion circuits, the CRT and I/O pads as well as the area of the reconfiguration elements in the PE. 6.1.1 Yield Estimates We will consider the case of no redundancy first, and then compare this to the reconfigured yield. For simplicity, assume average fatal defect densities of 1, 1.5 and 2.0 per cm2. Area values are based on a 0.8 fim CMOS technology (where A = 0.4 tim). The areas of the input and output conversion modules are based on floorplans consisting of cells developed and fabricated in the PE, and are A scalable to the 0.8 yum process. Thus, the area estimates presented are realistic, as they are based on existing cells. Table 6.1 depicts the areas of the major system components. It can be seen that the PE area dominates the bulk of the total system area (78.1%); therefore, this is the most logical place to apply fault tolerance. The yields for no redundancy are presented in Table 6.2, for various values of the clustering parameter a. Equation 3.26 has been used to obtain the values in Table 6.2. Let us now consider the case for the redundant sixteen processor array. The total area of each processing element is 0.008464 cm2, and the area of the switching elements in the PE is 0.002208 cm2 (ApE-n). The difference between these two areas (0.006256 cm2) represents the area of the PE which can tolerate a circuit fault (ApE-r). We have considered the case for one added redundant PE since this is the minimum amount of redundancy that can be added to the array, and since the switching out of more than one defective PE adds to the critical path delay as, each bypassed PE adds the delay of two series transmission gates plus interconnect delay .. Table 6.1: Table of System Component Areas. Areas of System Components Module Area Area Number Total Area [A x A] [cm2] [cm2] Input (Projected) Conversion: X, +x JX 1500 x 1500 0.0036 8 0.0288 y, -3y 1500 x 1500 0.0036 8 0.0288 Log 1150 x 750 0.0014 16 0.0224 Output (Projected) Conversion: 2-l(z + z*) 1550 x 800 0.0020 4 0.0080 2-1)-1(z z*) 2700 x 850 0.0037 4 0.0148 CRT 6800 x 5600 0.0609 2 0.1218 PE: (Actual) Normal 2300 x 2300 0.0085 64 0.544 Conjugate 2300 x 2300 0.0085 64 0.544 I/0 Non-Scalable Drivers (Actual) 0.0004 200 0.08 Chip II I 1.393 Table 6.2: Table of Non-Redundant Chip Yields. Projected Non-Redundant Chip Yields a Average Fatal Defect Densities per [cm2] 1.0 1.5 2.0 0.25 0.6246 0.5717 0.5357 0.50 0.5139 0.4394 0.3901 0.75 0.4550 0.3684 0.3125 1.00 0.4179 0.3237 0.2641 2.00 0.3474 0.2392 0.1746 .. (see Figure 4.2). As we will show shortly, just one extra processor will significantly improve the system yield. Equation 6.2 thus becomes: 17 17 17 17-MI 17-M2 17-Mg Y= E E E .E M=16M2=16 M8=16 k1=O k2=0 k8=0 (_l)kl(_l)k2 (_l)k8. 1 (17 )( 17M Ms 17 s [lI ((Mi + k1); + (M. + k8)K8 + AK) ] xl (6.5) Equation 6.5 was used in a computer program (see Appendix C) along with the previous values of defect densities and a, to produce Table 6.3. We see that the yield of the array changes from 62.46% to 72.53% in the best case (with Do = 1 per cm2 and a = 0.25), which is a 16.1% improvement. For the worst case (with Do = 2 per cm2 and a = 2), the yield changes from 17.46% to 35.87%, which is a 105.4% improvement. The added area overhead associated with the redundancy is _ 5%. Thus, this is a worthwhile tradeoff in both cases, and in particular for the worst case, in which the yield is doubled for a 5% increase in area. Equation 6.5 was extended to arrays of up to 32 processors. For the best case above, the yield of non-redundant vs. redundant is 55.04% and 67.33% respectively, which is a 22.32% increase. For the worst case, the results become 8.34% and 23.38%, a 180.3% increase. The area overhead for one redundant processor for the 32 PE case is P 2.7%, again, a worthwhile tradeoff. Any redundancy not used for yield enhancement at the time of manufacture, can be used for fault tolerance in the field, if a failure occurs in a PE. Since the PEs consist of 78% of the active area of our system, this will most often be the type of failure that occurs. .. Table 6.3: Table of Redundant Chip Yields. Projected Redundant Chip Yields a Average Fatal Defect Densities per [CM 2] 1.0 1.5 2.0 0.25 0.7253 0.6706 0.6318 0.50 0.6588 0.5798 0.5236 0.75 0.6259 0.5323 0.4657 1.00 0.6060 0.5028 0.4290 2.00 0.5703 0.4472 0.3587 Yield curves for the best and worst case manufacturing parameters have been plotted in Figure 6. L The top two curves represent the redundant and non-redundant best case, respectively. The bottom two curves represent the redundant and nonredundant worst case, respectively. At this point, some cautions are in order. We must be very careful when interpreting the increase in yield of redundant vs. non-redundant chips. Ultimately, all that really matters from a manufacturing standpoint is the number of good chips that leave the foundry. By including extra circuitry for redundancy, we have at the same time diminished the number of chips that are made. That is, the area of a redundant chip is larger than that of a non-redundant one, which translates into less chips per wafer. Since only good chips can be sold, we must be very careful to guarantee that the added redundancy does not cause too much product loss. If we are not careful, we could actually wind-up loosing money with redundant chips. In the following equations, we will represent the yield of non-redundant chips by YNR, and the yield of redundant chips by YR. The actual number of good chips for the non-redundant case will be denoted by KNR and for the redundant case, KR. The total number of chips fabricated for non-redundant and redundant cases is NNR and NR, respectively. Finally, the percentage of product loss, due to redundancy, will be denoted by PL. Consider the following: .. Yield Curves for Various Length Arrays 0.8 1 I 1 X . -. . .N X. .X. .X .N X. X X X . .'X X X. 0.7- --X x .x-.x.xxx.-. x -x -x.-x . 0.6- -. .- --.-. -OG . 0.5 0.4X . 0.3- x 0.2 0 0 . x . . x . . 0 . . 0 . . 0x . . x . . G . > . . . . . . . 0.1- .0 0. 0. 0 o. . .o.-. 0 0 .0.1 0 I I I I I I 16 18 20 22 24 26 28 30 32 Number of PEs Figure 6.1: Yield Curves for Various Length Arrays (Scheme 1). YNR = NNR YR = tA NR NR = (1- PL)NNR ? (6.6) KR > KNR 7 YR x NR > YNR X NNR YR x (1 PL) > YNR Equation 6.6 must always be satisfied to determine if the chosen redundancy scheme actually produced more working chips. We will assume that the percentage increase in area for redundant chips, translates into the same percentage of product loss. The degree of accuracy of this assumption ultimately depends on how many rectangular chips can fit into a round wafer. For the most part, though, this is a good .. assumption since a few hundred chips will typically fit into a commercially sized wafer (6-8 inch diameter). For our 16 processor example, there is thus a 5 % product loss. For the best and worst cases, we get (0.95) x best,, > YNRbest (0.95) x (0.7253) > 0.6246 0.6890 > 0.6246 (6.7) (0.95) X YR.o.,, > YNRwOTst (0.95) x (0.3587) > 0.1746 0.3407 > 0.1746 Thus, we obtain more working chips in each case, which supports the chosen redundancy scheme for the linear array. 6.2 A Comparison with Replacing Moduli An alternative redundancy scheme will be considered in this section. Suppose that fault tolerance was introduced into the system by including an additional modulus in the array, rather than an extra PE per modulus. This would have the effect of partitioning the system differently with respect to fault tolerance. The new "modules" would consist of a set of both normal and conjugate channels (for each modulus), together with the input and output conversion modules. This is depicted in Figure 4.1 by the dotted lines around each modulus. The CRT would now have to be reconfigurable, as the dynamic range of the system would change when a defective module is switched out and a spare switched in. If the system were configured in this way, then faults could be tolerated in the input and output conversion modules, as well as in the PEs. The only "chip-kill" area, is that composed of the CRT (which now is larger) and the 1/0 pads. The system would now consist of a single .. "module-type", of which there are four total for the non-redundant case and five for the redundant. There are now five fixable combinations, compared to eight as before. Since there is only a single module type, Equation 6.2 reduces to: N1 N, -M, M1=N1 -RI kl=0 )k N 1 N1 M1 )X[l ((MI + kj)AX + ACK) (-1) M x 1 + X CM (6.8) For this scheme, the corresponding parameters are: N, = 5 R1 =1 A1 = 2 x (APEr + APE-) 16 + 4 x (Ajconl + ALogl) + Aoutconvl ACK = AIOpads + ACRTN, (6.9) with CM = 1 for M E {4, 5}. We have divided the area of the PE interconnections by two since we do not bypass PEs in this scheme, and thus, only need half of the wires. A,Co,v,, is the area of one forward QRNS mapping module and ALog1 is the area of a single log table. There are four of each of these per reconfigurable module. There is one inverse QRNS block per reconfigurable module, the area of which is denoted by Aoutco,n,. We will assume the I/O overhead is the same as before. Finally, we need to obtain a value for the area of the reconfigurable CRT. Approximately 2 of the CRT is composed of ROMs. Since the new CRT will have to be re-programmable (i.e. to change the overall M), this section will need to be .. RAM based. RAMs occupy about four to six times the area of ROMs (as stated earlier). We will use a value of four as a multiplication factor, to estimate the area of a RAM based implementation. The carry-select mod(M) adders used in the CRT will also have to be reconfigurable. We will not be able to use the bit-programming scheme described in Chapter Four, and thus, will increase the complexity by a factor of 25 %. The new variable offsets will have to be stored in registers also. We will again be very generous and assume that the total complexity of the modulo adder portion of the CRT increases by 50 %. The overall area multiplication factor for the reconfigurable CRT is 3.166. If we substitute the redundancy parameters defined above in Equation 6.8 the following expression results: 5 5-M, r= E E M1=4 k,=O (1)ki ( ) Mi) X[1 ((M + &kl)xl+ACK CM, (6.10) The area values for the new system were substituted in Equation 6.10, which was evaluated with a computer program (see Appendix C). For our sixteen processor example, the best case non-redundant vs. redundant yield is 60.80 % vs. 69.27 %. For the worst case manufacturing parameters, the yields are 15.04 % vs. 26.18 %. How do these values now favor with product loss accounted for? The overhead associated with this redundancy scheme is approximately 41 %. Thus, after scaling the redundant yields by unity minus this amount we get- (1 0.41) x 69.27 = 40.87 < 60.80 and for the worst case (1 -0.41) x 26.18 = 15.44 > 15.04. For the best case of manufacturing parameters we have actually lost money by including redundancy. That is we would have obtained more working chips if we had done nothing! For the worst case, the gains are questionable since the margins are so low. The yield curves for best and .. worst cases are plotted in Figure 6.2 for 16 to 32 processor systems. The areas of chips using the first and second redundancy schemes are plotted in Figure 6.3. We point out that the areas of chips using the first scheme are less than those using the second, thus the first scheme always wins. Finally the product loss adjusted yields are plotted for both schemes for best and worst cases. Figure 6.4 and Figure 6.5 are plots for the best and worst cases, respectively for the first scheme. These curves illustrate that we always benefit for this scheme, for these manufacturing parameters. Figures 6.6 and Figure 6.7 show the curves for the second scheme. Figure 6.6 shows that we always loose money for this case, while Figure 6.7 shows that as the array size grows, we gain more for the implemented redundancy. This is because the percentarea-overhead associated with the redundancy decreases as the array grows (in both schemes). The results of this analysis suggests that it is better to implement redundancy within the moduli, rather than between the moduli. Thus, the redundancy scheme used in the proposed system is justifiable. In reconfigurable systems, the amount of redundancy must be kept as small as possible to obtain the most benefit. In much of the RNS literature, the point is commonly made that fault tolerance can be achieved by including extra moduli. However, in the context of this analysis, this point is questionable. Finally, it should be pointed out that if the moduli are very small, such that the inclusion of extra channels will not significantly increase the chip area, then the analysis may be more favorable. 6.3 Detecting Faults The detection of failed processors in the chosen scheme must be done off-line. The test procedure combines the bypassing of failed PEs to isolate columns in which failed PEs lie in and then suppling specific test vectors to determine which row of the column has failed. For example, all columns in the array can be bypassed except .. |

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xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0008238600001datestamp 2009-02-16setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title A fault tolerant GEQRNS processing element for linear systolic array DSP applicationsdc:creator Smith, Jeremy C.,dc:publisher Jeremy C. Smithdc:date 1994dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00082386&v=0000132490064 (oclc)002007920 (alephbibnum)dc:source University of Floridadc:language English
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